The Tangential Cauchy Reimann Equations & CR Manifolds

CR Manifolds and the Tangential Cauchy- Riemann Complex

STUDIES IN ADVANCED

TWS

Studies in Advanced Mathematics

CR Manifolds and the Tangential Cauchy—Riemann Complex

Studies in Advanced Mathematics

Series Editor Steven G. Krantz Washington University in S Louis

Editorial Board R. Michael Beals

Gerald B. Folland

Rutgers University

University of Washington

Dennis de Turck

William Helton

University of Pennsylvania

University of California at San Diego

Ronald DeVore

Norberto Salinas

University of South Carolina

University of Kansas

L. Craig Evans

Michael E. Taylor

University of California at Berkeley

University of North Carolina

Volumes in the Series Real Analysis and Foundations, Steven G. Krantz CR Manifolds and the Tangential Cauchy—Riemann Complex, Albert Boggess Elementary Introduction to the Theory of Pseudodifferential Operators,

Xavier Saint Raymond Fast

Fourier Transforms, James

S.

Walker

Measure Theory and Fine Properties of Functions. L. Craig Evans and

Ronald Gariepv

ALBERT BOGGESS Texas A & M University

CR Manifolds and the Tangential

Cauchy—Rieman n Complex

CRC PRESS Boca Raton Ann Arbor

Boston

London

Library of Congress Cataloging.in-Pubhcation Data Catalog record is available from the Library of Congress

This book represents information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Every reasonable effort has been made to give reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. All rights reserved. This book, or any parts thereof, may not be reproduced in any form without written consent from the publisher.

by Archetype Publishing Inc., P.O. Bo' 6567,

This book was formatted with Champaign, IL 61821.

Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida, 33431.

©

1991

by CRC PreSS, Inc.

International Standard Book Number 0-8493-71 52-X

Printed in the United States ofAmerica

1 234567890

Contents

Contents

Introduction PART I. PRELiMINARiES

1

1

Analysis on Euclidean Space

2

1.1

Functions Vectors and vector fields Forms The exterior derivative Contractions

2

17

2.2 2.3 2.4 2.5

Analysis on Manifolds Manifolds Submanifolds Vectors on manifolds Forms on manifolds Integration on manift)lds

3

Complexijied Vectors and Forms

39

3.1

Complexification of a real vector space

39

1.2 1.3 1.4 1.5

2 2.1

17 19

23

26 29

Contents

3.2 3.3

Complex structures Higher degree complexified forms

4

The

Frobenius Theorem

5/

4.1

The real Frobenius theorem The analytic Frobenius theorem Almost complex structures

51

61

5.1

Distribution Theory The spaces 7)' and e'

5.2 5.3 5.4

Operations with distributions Whitney's extension theorem Fundamental solutions for partial differential equations

65

6

Currents Definitions Operations with cunents

79

PART II: CR MANIFOLDS

95

7

CR Manifolds

97

7.1

Imbedded CR manifolds A normal form for a generic CR submanifold Quadric submanifolds Abstract CR manifolds

4.2 4.3

5

6.1

6.2

7.2 7.3 7.4

8 8.1

8.2 8.3

The Tangential Cauchy—Riemann Complex Extrinsic approach Intrinsic approach to C9M

41

45

56 58

61

71

74

79 84

97 103 111

120

122 122

130

The equivalence of the extrinsic and intrinsic tangential Cauchy—Riemann complexes

134

9

CR

9.1

CR functions CR maps

140

The Levi Form Definitions

156

9.2

10 10.1 10.2

Functions and Maps

The Levi form for an imbedded CR manifold

140 149

156 159

ix

Contents

The Levi form of a real hypersurface

The Imbeddability of CR Manifolds The real analytic imbedding theorem Nirenberg 's nonimbeddable example

12

Further Results

12.1 12.2 12.3 12.4 12.5

Bloom—Graham normal form Rigid and semirigid submanifolds

More on the Levi form Kuranishi 's imbedding theorem Nongeneric and non-CR manifolds

163

169 169 172

179 179 183 185 187 187

PART 1!!: THE HOLOMORPHIC EXTENSION OF CR FUNCTiONS

189

An Approximation Theorem

191

14

The

14.1

14.2 14.3

Lewy's CR extension theorem for hypersurfaces The CR extension theorem for higher codimension Examples

15

The

15.1 15.2 15.3

15.4 15.5

Reduction to analytic discs Analytic discs for hypersurfaces Analytic discs for quadric submanifolds Bishop's equation The proof of the analytic disc theorem for the general case

16

The

16.1

A Fourier inversion formula The hypoanalytic wave front set The hypoanalytic wave front set and the Levi form

230 237 244

Further Results The Fourier integral approach in the nonrigid case The holomorphic extension of CR distributions CR extension near points of higher type

251

16.2 16.3

17 17.1 17.2 17.3

Statement of the CR Extension Theorem

Analytic Disc Technique

Fourier Transform Technique

198 198

200 202

206 207 208 210 214 221

229

251

254 257

x

17.4

Contents

Analytic hypoellipticity

PART IV: SOLVABILITY OF THE TANGENTIAL CAUCHY-R1EMANN COMPLEX

263

18

Kernel Calculus

265

18.1

18.2

Definitions A homotopy formula

265 272

19

Fundamental Solutions for the Exterior Derivative

and Cauchy—Riemann Operators

277 278

19.2 19.3

Fundamental solutions for d on Fundamental solutions for 0 on Bochner's global CR extension theorem

20

The

20.1

20.4

A general class of kernels A formal identity The solution to the Cauchy—Riemann equations on a convex domain Boundary value results for Henkin's kernels

21

Fundamental Solutions for the Tangential

19.1

20.2 20.3

Kernels of Henkin

Cauchy—Riemann Complex on a Convex Hypersuiface 21.1

A

23.1

23.2

299 303

312 312 317

Local Solution to the Tangential

Cauchy—Riemann Equations

23

294 297

A second fundamental solution to the tangential Cauchy—Riemann complex

22

294

The first fundamental solution for the tangential Cauchy—Riemann complex

21.2

281 291

Local Nonsolvability of the Tangential Cauchy—Riemann Complex Hans Lewy's nonsolvability example Henkin's criterion for local solvability at the top degree

327

334 334 337

xi

Contents

24

Further Results

342

24.1

More on the Bochner—Martinelli kernel Kernels for strictly pseudoconvex boundaries Further estimates on the solution to Wealdy convex boundaries Solvability of the tangential Cauchy—Riemann complex in other geometries

342 345 348 348

Bibliography

354

Notation

359

Index

361

24.2 24.3 24.4 24.5

349

Introduction

theory of complex manifolds dates back many decades so that its origins are considered classical even by the standards of mathematicians. Consequently, there are many fine references on this subject. By contrast, the origins of the theory of CR manifolds are much more recent even though this class of manifolds contains very natural objects of mathematical study (for example, real hypersurfaces in complex Euclidean space). The first formal definition of The

the tangential Cauchy—Riemann complex did not appear until the mid 1 960s with the work of Kohn and Rossi [KR]. Since then, CR manifolds and the tangential Cauchy—Riemann complex have been extensively studied both for their intrinsic

interest and because of their application to other fields of study such as partial differential equations and mathematical physics. The purpose of this book is to define CR manifolds and the associated tangential Cauchy—Riemann complex and to discuss some of their basic properties. In addition, we shall sample some of the important recent developments in the field (up to the early 1980s). In the last two decades, research on the subject of CR manifolds has branched into many areas. Two of these areas that are of interest to us are (I) the holomorphic extension of CR functions (solutions to the homogeneous tangential Cauchy—Riemann equations) and (2) the local solvability or nonsoivability of the tangential Cauchy—Riemann complex. The first area started in the 1950s when Hans Lewy [Li] showed that under certain convexity assumptions on a CR functions locally extend to holomorphic functions. real hypersurface in Over the years, many refinements have been made to this CR extension theorem so that it now includes manifolds of higher codimension with weaker convexity

assumptions. The second area started in the i960s with the work of Kohn. He used a Hilbert space (L2) approach to construct solutions to the tangential Cauchy—Riemann complex on the boundary of a strictly pseudoconvex domain (except at top degree). Later, Henkin developed integral kernels to represent solutions to the tangential Cauchy—Riemann equations. A closely related topic is the nonsolvability of certain systems of partial differential equations. In the 1950s, Hans Lewy constructed an example of a partial differential equation with smooth coefficients that has no locally defined smooth solution. In particular, he showed that cannot replace "real analytic" in

xiu

xiv

the statement of the Cauchy—Kowalevsky theorem. Lewy 's example is closely related to the tangential Cauchy—Riemann equations on the Heisenberg group in C2. His example illustrates that the tangential Cauchy—Riemann complex on a strictly pseudoconvex boundary is not always solvable at the top degree. Later, Henkin developed a criterion for solvability of the tangential Cauchy—Riemann complex at the top degree. The first half of this book contains general information on the subject of CR manifolds (Part II) and the prerequisites from real and complex analysis (Part I). In Parts III and IV, we develop the subjects of CR extension and the solvability of the tangential Cauchy—Riemann complex. This book is not a treatise. We do not discuss the £2-approach to the global solvability of the tangential Cauchy—Riemann equations. This material is contained in Folland and Kohn's book [FK] and our work certainly could not offer any improvements. Instead, the integral kernel approach of Henkin is presented. The local theory dealing with points of higher type (points where the Levi form vanishes) is not presented in detail. The theory of points of higher type is too immature or too complicated for inclusion in a book at this time. Instead, the end of each pan contains a chapter entitled Further Results where some of the recent literature on function theory near points of higher type and other topics are surveyed with few proofs given. Each writer has his own peculiar style and tastes and this author is no different. The reader will notice that I favor the concrete over the abstract. This may offend some of the purists in the audience but I offer no apologies. I firmly believe that a student learns much more by getting his or her hands dirty with some analysis rather than by merely manipulating abstract symbols. In this book, abstract concepts are introduced after some motivation with the concrete situation. It is hoped that the audience for this book will include researchers in several complex variables and partial differential equations along with graduate students who are beyond their first year or two of graduate study. The reader should be familiar with advanced calculus, real and complex analysis, and a little functional analysis (at least enough so that he or she does not faint at the sight of a Banach or Fréchet space). Although this book cannot reinvent the wheel, many of the prerequisites for reading Parts II through IV are given in Part I. We start with a discussion of vectors and forms, both in the Euclidean and manifold setting. A proof of Stokes' theorem is given since it is such a basic tool used throughout the book. Proofs of the smooth and real analytic versions of the Frobenius theorem

are given since these theorems are used in the imbedding of CR manifolds. At the end of Part I, a rapid course in the theory of distributions and currents is given. This material will be essential in Part IV. There are other elementary topics that are not included in Part I. These include the existence and uniqueness theorem for ordinary differential equations and the Cauchy—Kowalevsky theorem for partial differential equations. Even though these topics are no more advanced than other topics covered in Part I (for example, Stokes' theorem), they are not as frequently used in this book and therefore we only give references. Surprisingly

xv

little theory from several complex variables is used. For most of the book, the reader needs only to be familiar with the definition and basic properties of holomorphic functions of several complex variables. However, the book will certainly be more meaningful to someone who has a further background in several complex variables. The Newlander—Nirenberg theorem is the most advanced topic from several complex variables that is used in the book. It is only used in the discussion of Levi flat CR manifolds. A proof of the imbedded version of the Newlander—Nirenberg theorem is provided for the reader who wishes to restrict his or her attention to imbedded Levi flat CR manifolds. Part II covers general information about CR manifolds and the associated tangential Cauchy—Riemann complex. We start with the definitions of imbedded

and abstract CR manifolds. In addition, we present a normal form that gives a convenient description of an imbedded CR manifold in local coordinates. Next, we introduce the tangential Cauchy—Riemann complex. For an imbedded CR manifold, an extrinsic approach is given that makes use of the ambient complex For an abstract CR manifold, an intrinsic approach is given that structure on makes no use of any ambient complex structure (since none exists). In the case of an imbedded CR manifold, these two approaches are technically different, but we show they are isomorphic. Our approach to the tangential Cauchy— Riemann complex makes use of a Hermitian metric. We also mention a more invariant definition of the tangential Cauchy—Riemann complex that does make use of a metric, but this approach is not emphasized because calculations usually require a choice of a metric. CR functions and CR maps are then introduced. We prove Tomassini's theorem [Tom], which states that a real analytic CR function on an imbedded real analytic CR manifold is locally the restriction of an ambiently defined holomorphic function. This theorem does not hold for the class of smooth CR functions. However, we do show that a smooth CR function is always the restriction of an ambiently defined function that satisfies the Cauchy—Riemann equations to infinite order on the given CR submanifold. Next, we introduce the Levi form, which is the complex analysis version of the second fundamental form from differential geometry. An extensive analysis of the Levi form for the case of a hypersurface is given. In particular, we show the relationship between the Levi form and the second fundamental form. We then show that any real analytic CR manifold can be locally imbedded as a CR submanifold of The smooth version of this theorem is false in view of Nirenberg's counter example [Nir], which is also given. In the chapter entitled Further Results we discuss some related results such as the Bloom— Graham normal form for CR submanifolds of higher type, rigid and semi-rigid CR structures, and Kuranishi's imbedding theorem. Most of these results are presented without proof. Part III discusses the local holomorphic extension of CR functions from an imbedded CR manifold. We start with an approximation theorem of Baouendi and Treves [BT 1], which states that CR functions can be locally approximated by entire functions. Their theorem is more general but we restrict our focus to CR

xvx

functions to simplify the proof. Next, we state the CR extension theorem, which is a generalization of Hans Lewy's hypersurface theorem alluded to above. In addition, the convexity assumptions of this theorem are discussed and examples are given. We present two techniques for the proof of this theorem. Both of these techniques are used in today's research problems and thus these techniques are as important as the CR extension theorem. The first technique involves the use of analytic discs and it was originally developed by Lewy [Li] and Bishop [Bil. This technique together with the approximation theorem yields an easy proof of Hans Lewy's CR extension theorem for hypersurfaces. An explicit proof is also given in the case of a quadric submanifold of higher codimension. The proof for the general case requires an analysis of the solution of a nonlinear integral equation involving the Hilbert Transform (Bishop's equation). The second, more recent, technique involves a modified Fourier transform approach due to Baouendi and Treves. The idea here is to obtain the holomorphic extension of a given CR function from the Fourier inversion formula — suitably modified for CR manifolds. This technique is applicable to CR distributions and points of higher type. However, to avoid technical complications, we give the details of this technique only for the case of smooth CR functions on a type two CR submanifold. Some of the extensions of this technique to CR distributions are discussed at the end of Part III in the chapter entitled Further Results. Part IV deals with the solvability and nonsolvability of the tangential Cauchy— Riemann complex on a strictly pseudoconvex hypersurface in The approach taken here involves Henkin's integral kernels although we use the notation and kernel calculus set down by Harvey and Polking [HPI. We give two fundamental solutions for the Cauchy—Riemann complex on — the Bochner—Martinelli

kernel [Boc] and the Cauchy kernel on a slice. These kernels together with the kernels of Henkin yield a solution to the Cauchy—Riemann equations on a strictly convex domain in Furthermore, these kernels provide an easy proof of Bochner's theorem, which states that a CR function on the boundary of a bounded domain with smooth boundary globally extends to the inside as a holomorphic function. Next, a global integral kernel solution for the tangential Cauchy—Riemann complex is given for a strictly convex hypersurface. These kernels are then modified to yield Henkin's [He31 local solution to the tangential Cauchy—Riemann equations. We then present Henkin's criteria for local solvability of the tangential Cauchy—Riemann complex at the top degree. Results on the solvability of the tangential Cauchy—Riemann complex on hypersurfaces under other geometric hypotheses are given in the chapter entitled Further Results. My point of view in mathematics has been influenced by a number of people whom I have the pleasure to thank. First, I owe a lot to my thesis advisor, John Polking. He along with Reese Harvey has shaped my mathematical development since my early graduate school years at Rice University. Even though I have

never met him, 0. M. Henkin has provided a lot of inspiration for much of my work. Other mathematicians who have influenced my mathematical point

xvii

of view include Salah Baouendi, Al Taylor, Dan Burns, and Alex Nagel. The reviewers did an excellent job of finding errors and making helpful suggestions. I also wish to thank Steve Krantz (who initially encouraged me to write this book) and the rest of the editorial staff at CRC Press for having the confidence in me to complete this project. I wish to thank Texas A&M for support during the preparation of most of this project. I did the final editing while visiting Colorado College and I want to thank their mathematics department for their hospitality during my visit.

In addition, I wish to thank Robin Campbell, who typed portions of this manuscript and answered many of my questions concerning I also wish to thank my son for putting up with me during the preparation of this manuscript. Finally, I wish to thank Steve Daniel and the rest of the Aggieland Paddle club for convincing me that from time to time, I need a break from the book writing to go whitewater kayaking.

Al Boggess, October 1990 Colorado Springs, CO

Part I Preliminaries

In this first part, we provide most of the prerequisites for reading the rest of the book. We start with a review of certain aspects of function theory, vectors, vector fields, and differential fonns on Euclidean space. These concepts are then defined in the context of manifolds. Proofs are given for Stokes' theorem and its corollaries — Green's formula and the divergence theorem. A proof of the Frobenius theorem is then given. The real analytic version of this theprem is also discussed since it will be used for the imbedding theorem for real analytic CR manifolds in Part II. We discuss distribution theory as applied to partial differential equations. Fundamental solutions for the Laplacian on and the Cauchy—Riemann equations in one complex variable are given. They are used in Part IV, where we discuss fundamental solutions to the Cauchy— Riemann equations on and their analogue on a real hypersurface of — the tangential Cauchy—Riemann equations. These systems of partial differential equations act on differential forms. Therefore we shall need a distribution theory for differential forms, i.e., the theory of currents. This and related topics are reviewed at the end of Part I. Excellent references are available for all of the topics in Part I. These include Spivak's volumes on differential geometry [Sp], Krantz's book [Kr] or Hörmander's [Ho] for several complex variables, Yosida's book [Y] for functional analysis and distribution theory (see also Schwartz [Sch}), Federer's book [Fe] for geometric measure theory, and John's [Jo] or Folland's book [Fo] for partial differential equations.

1 Analysis on Euclidean Space

Here, we discuss some function theory and define the notions of vectors and forms on Euclidean space.

1.1

Functions

There are several classes of functions we shall use. For an open set 11 in let CIc(1l) = the space of k-times continuously differentiable real- or

complex-valued functions on

=

space of infinitely differentiable real- or complexvalued functions on ci, the

= the space of elements of compact support.

with

We shall make use of a special class of mollifier functions {x€; V(ci). This class is defined as follows. Let

for x E

with IxI

> O} C

1

and let

= tx Each

2

is smooth. Here,

.

IIcI(RN) denotes the usual C'-norm of a function.

Functions

3

The following properties can easily be shown: (i)

lxi

(ii) JXEE' x(x)dx = 1. These two properties allow the construction of cutoff functions as described in the following lemma. LEMMA 1

Given a compact subset K of an open set Cl C to V(fl) with 1 on a neighborhood of K.

there is function

belonging

PROOF We first choose a compact set K1 C Cl so that K is contained in the interior of K1. Let

ifxEKi

I I = o

'K1 (x)

x

Choose c > 0 small enough so that 2€ is less than the smaller of the distance between K and Cl — K1 and the distance between K1 and RN — ci Let *

Here, * is the usual convolution operator in RN, so

J x€(s—y)1K1(y)dy. yERN

is smooth, clearly is smooth. Property (1) for xe shows that vanishes outside an c-neighborhood of K1. So 1j has compact support in Cl by our choice of e. Property (ii) for xf and our choice of e imply that q5 1 on an c-neighborhood of K. Therefore, is our desired cutoff function. I Since

As an immediate corollary, we can construct partitions of unity, as described in the following lemma. LEMMA 2 Suppose

U3 is an open subset of

set in Ri" with K C Ui U ...

U

for j 1,. . , m. Suppose K is a compact Urn. Then there is a collection of functions .

such that (i)

1

on a neighborhood of K.

Often, the collection cover {U3}.

is called a partition of unity subordinate to the

Analysis on Euclidean Space

4

PROOF

First, we find open subsets Uj',... ,

with

C U3 so that K C

E D(U,)

Next; we choose cutoff functions a neighborhood of LI3. Then let U

U

1 on

with ii-',

.0(x)

where

with 0

E

easily show that the set

}

1

on a neighborhood of K. The reader can

satisfies the conclusions of the lemma.

I

The key idea in the proof of Lemma 1 is that the characteristic function for a compact set K (IK) can be approximated by the sequence of smooth functions x satisfies properties (i) and (ii) listed just before Lemma 1. Convolving with x€ can be used to approximate other types of functions as well. For example, if I is a continuous function on RN, then properties (i) and (ii) can be used to show that the smooth sequence * 1) converges to f uniformly on compact sets as e —' 0. { Another important class of functions is

=

space of real analytic functions on real or complex valued).

(either

the

A function f is real analytic

on

f

an open set

if in a neighborhood of each

point in can be represented as a convergent power series. It is a standard fact that A(11) is a subset of e(IZ), i.e., real analytic functions are smooth. is given by the following. Let A real analytic version of the

e(x) = The

power series for e(x)

ii._N/2e_1x12

about the

origin

for

x

E

converges

for all

x,

so e belongs

to

For€>0,

€.

e set of functions to (i) and (ii) for

The

{

} satisfies the

following properties

(i) Given t5 > 0,

f

which are analogous

0.

IvI6

(ii)

JRN eEQJ)dy

=

1, for each

0.

follow from a standard polar coordinate calculation after the change of variable t = y/€. we now prove the classical Weierstrass theorem, which states Using the that is a dense subset of 6(e) in the following topology for E(11): a These properties

Functions

5

is said to converge to f in sequence f,,, E compact subset K of and for each multiindex negative integer)

D°f

as n = (ni,...

uniformly on K as n —p

,

00 if for each (o, a non-

00.

Here, and

= THEOREM

1

.. .

lal

=a+

+ aN.

WEIERSTRASS

Suppose f belongs to Then there is a sequence of polynomials P1, P2,. that converges to f in the topology of e(11).

Let K be an arbitrary compact subset of 11 and let E be a cutoff function that is identically 1 on a neighborhood of K. If I belongs to then belongs to V(1l). For > 0, define PROOF

F€(x) =

= =

*

J

YERN

J yEW"

—

y)f(x

—

y)dy

(by a translation).

Note that

=

J yEW"

—

y)dy.

In view of property (ii) for

=

=

Therefore, for x E K (so

-

=

f

J

yEW" 1),

we have

- y) -

YERN

For 6 > 0, we split the integral on the right into the sum of an integral over

6} and an integral over 6). The first of these integrals can be With made small by choosing 6 small using the uniform continuity of this choice of 6, the second of these integrals converges to zero as —p 0 in uniformly on converges to view of property (1) for eE. Therefore,

K as

0.

Analysis on Euclidean Space

6

The power series for each FE about because F( can be written

FE(x)=CN

origin converges

the

for all

x E Ri"

J

yEE'

the power series for e(.) can be integrated term by term. By truncating the power series of F( about the origin, we obtain a sequence of polynomials n = 1,2,... such that for each multiindex and because

sup

—

D°f(x)I

0

as

n —p

xE K

Now let K1, K2,... be an increasing sequence of compact sets with Let be a polynomial with sup

Clearly,

!.

—

f in E(ci) as n —p

and

= ci.

the proof of the theorem is complete. U

The final class of functions to consider is the class of holomorphic functions.

If ci is an open set in C'2, then let

O(ci) = the space of holomorphic functions on ci. A function is holomorphic on ci if it satisfies the Cauchy—Riemann equations on

ci,

where

.0 0z3 Here,

2 \0x3

Oy,

we have labeled the coordinates for C'2 as (z1,... ,

with z3

x2 +

(i = We assume the reader knows some basic complex analysis. If I is holomor. then the reader should know phic in a neighborhood of a point p that f can be expressed as a convergent series in powers of (z1 —Pi), . .. , — pa). This and a connectedness argument imply the identity theorem for holomorphic functions: if f is holomorphic on a connected open set ci and if f vanishes on an open subset of ci, then f vanishes everywhere on ci. This is often expressed by saying that an open set is a uniqueness set for holomorphic .

.

Vectors and vector fieWs

7

functions. Other types of sets are also uniqueness sets for holomorphic func-

tions. For example, if f is holomorphic on and if I vanishes on the copy of given by {(x1 + iO,. .. + iO);(x1,. .. then f vanishes E everywhere. This is because all the v-derivatives of f vanish on this copy of The Cauchy—Riemann equations can then be used to inductively show that all x, y derivatives of I vanish on this copy of In particular, a power series expansion of f about the origin must vanish identically and hence f 0. —' Ctm is called a holomorphic map if each A function f = (f',.. . , fm): I,: C is holomorphic, I <j <m. This definition can be recast in terms of the real derivative of f (denoted Df) as a map from to R2m. Define the complex structure map

J:

—.

by

By the Cauchy—Riemann equations, a C' map f: Il if and only if for each point p E

—, Ctm

is holomorphic in

Df(p)oJ=JoDf(p) as linear maps from to R2m. The J on the left is the complex structure map for whereas the J on the right is the complex structure map for R2m.

1.2

Vectors and vector fields

Now, we turn to the topic of vectors and vector fields on RN. These concepts will be generalized to the manifold setting in Chapter 2. A vector at a point p E RN is an operator of the form

v,ER. The set of vectors at p is denoted (RN) and is called the (real) tangent space of RN at p. An element L in can be viewed as a linear map L: ((RN) —. R by defining

L{f}

=

Analysis on Euclidean Space

8

A vector field on an open set Il C is a smooth map that assigns to each point p E ci a vector in TP(RN). Here, the concept of smooth means that each vector field L can be written

L= where each v3 is an element of S(ci). We let

be the value of the vector field

L at p, i.e.,

= We also write

(Lf)(p) =

=

(p E ci),

Thus, Lf is also an element of e(ci). RN' is a smooth (C') map near a point p E RN. F Now suppose F: R1" TF(p)(RN') called the push forward of F at p. induces a map F.: It is defined by

for f E e(rz).

F.(L){g} = L{goF} for L E

and g E e(RN'). If L is a vector field on Il, then the above

equation reads

= for g E e(RN') and p E ci. If F : ci '—' Il' is a diffeomorphism, then F.(L) is a vector field defined on ci'. Let us compute F.(O/0x3), I <j
f8\ {xk} =

8 OFk (9X3

where

we have written F = (F,,... ,FN'). Therefore

[F.

Forms

9

If L =

v30/0x3 belongs to TP(RN), then

[F,(L)]F(P) =

(_)]

N' /j%t

/_. Note that the coefficients of [F. (L)]

are

tive matrix DF(p) with the vector (v1,

,

obtained by multiplying the derivaVN).

Forms

1.3

The dual space of TP(RN) is denoted T (RN) and this is called the space of 1-forms on RN at p. By definition, this is the space of all linear functionals on If is a 1-form and L is a vector at p then we denote the value of o L by

on

0

\ if

1 <j
1 <j
Let

=

\

f

ifj=k

1

Jk_l 0

a 1-form and L =

vk(O/Oxk)

is

a vector, then

= Note that (ci, L) is the same as the Euclidean inner product between (cki,...

(V1,...,VN) as elements of RN. A differential 1-form on an open set

,

and

C RN is an object of the form

= where each

is an element of 6(e). An important example of differential

1-form is the exterior derivative of a function f, denoted df. This is defined as and L E let follows. For I E p

= L{f}.

Analysis on Euclidean Space

10

a coordinate function x3 and L be the vector O/ôxk, we see that the exterior derivative of x3 is the same as the dual basis definition of dx3 given above. By applying this definition of df to the vector O/Oxk, we have

Letting /

be

df(p) = k=1

OSk

be the rth exterior power of V. A

If V is a vector space, then we let basis of is given by

l<ji<"<jr
.

vA(au+&w)=a(vAu)+b(vAw)

a,bER

(au + bw) A v = a(u A v) + b(w A v)

v, u, w E V

and anti-symmetric, i.e.,

vAw=—wAv

v,wEV.

In particular, v A v = 0. In the context of vectors and forms, we have the space of r-vectors A basis for each is given by and the space of r-forms

ía

a

(RN))

a

ax and

{dx' =dx1 A...Adxjr;

1


i1

can be viewed as the space of linear functionals on

The space the space A

A

... A Lr)p =

T(RN) and L1 all permutations a of the set {l,.

La(i))p...

La(r))p

and where the sum runs over the set of , r}. Here, sgn(a) is the factor +1 if the . permutation a is even and —1 if the permutation is odd. The collection {dx' } is a dual basis for i.e., for

\ If F:

—'

can

X

.

8 \ _f 1 ifl=J —

0

otherwise.

is a C' map then the push forward map be extended to a linear map

Forms

11

by defining

F.(L1A . .. A Lr) = F.(L1)A ... A F.(Lr). _+

can dualize this map to obtain the pull back map F: For r = 1, the pull back is defined by

We

=

for

TP(RN).

0

For r 1, define for

The equation

(F(ØI)A. ..AF(Or),L1A. .ALr>p = .

..AF.(Lr))p(p)

.

follows easily from the above definitions. It is useful to know how to compute the pull back of a form using coordinates.

We start with the 1-form dx,. From the definitions, we have

t9xk

F(p)

Oxj

11 OF,(

F'(p)

(/Xk So

at any point p E

we have

(Fdx, ) (p) =

dxk

= If

= Qdx1

A... A

e R), then

by the definitions and the above computations. If is a smooth function, then on the right, must be composed with F, because the pairing on the right side is occurring at the point F(p). of the definition (F'O, = We have

A...AdFI,. for

Analysis on Euclidean Space

12

The pull back of the volume form dx = dx1 A ... out as a special case. Suppose N = N'. We have

A

dXN should be singled

F*dX=dFIA...AdFN. By expanding each dF, and using the multilinearity of the wedge product, we obtain

Fdx = (detDF)dx.

1.4

The exterior derivative

We have already defined the exterior derivative of a smooth function and we have shown for

fE6(RN).

j=I In this section, we extend the definition of the exterior derivative to higher degree forms and then discuss some of its properties. form of degree r, or more First, we require a definition. A smooth simply, a smooth r-form on an open set C IRN is an object of the form III=r

The sum is over all increasing indices I is an element of The space of all smooth r-forms on Q is denoted e'(cl). of length = = r. the space of smooth r-forms with compact support. We denote by V' where each

DEFINITION 1

The exterior derivative d:

(111=r

Since

III=r

)

df1 is a smooth 1-form, the right side is a smooth (r + 1)-form.

LEMMA I (i) (product rule)

(ii)

is defined by

d2 =

0, i.e.,

A

E

=

A

+ (—

A

for 4i E

then d(dçb) = 0.

Property (i) follows from the product rule for differentiation. From the definition of the exterior derivative, it suffices to show property (ii) for functions PROOF

The exterior derivatIve

13

(0-forms). This follows by an easy computation using the fact that the mixed partial derivatives of a smooth function are independent of the order in which they are taken. I

er+l(cl); 0

Property (ii) is often described by stating that {d:

r N} forms a complex. The exterior derivative commutes with pull back, as the following lemma shows. LEMMA 2 Suppose F* o d =

C RN and Il' C RN' and suppose F: il' is a C' map. Then do F as maps from to er+l (Il) for 0 < r N'.

Suppose 0 = adx1 with a E £(11') and for F*0 at the end of the previous section, we have PROOF

= r. From the formula

F0 = (a o F)dF'. Since d2 = 0, clearly d(F*O) = d(a o F) A dF'. On the other hand,

Fd4 = F*(da A dx') = F(da) A dF'. Comparing these two expressions, we see that the lemma will follow provided we show

F*(da)=d(aoF)

for

a€e(&T).

Using the notation F (a) = a o F for a function a, this equation can be viewed as the statement of the lemma for the case r = 0. This can be proved by using the chain rule.

I

I

In the previous section, we defined the exterior derivative of a smooth function by its action on a vector field X, i.e.,

= There is a corresponding formula that relates the exterior derivative of a higher

degree form in terms of its action on a wedge product of vector fields. The

Analysis on Eucll4ean Space

14

formula is rather messy. Moreover, we shall only need this relationship for the exterior derivative of 1-forms. We present this case in the following lemma. LEMMA 3

Suppose

is a smooth 1-form and L', L2 are smooth vector fields. Then (d4., L' A L2) = L'{(4, L2)} — [L',L2]).

L')}

Some explanation is in order. First, 0 is a 1-form; so the pairings L') and (0,L2) are smooth functions. Therefore, L2{(qS,L')} and L'{(0,L2)} are the actions of the vector fields L2 and L' on the smooth functions L') and L2). The notation [L', L2] in the third term on the right is the Lie bracket of the vector fields L1 and L2. At a point p RN, the Lie bracket is defined by

An easy computation using the coordinate functions f = x3, that if

1

j

N,

shows

then

[L',L2] =

—

We first note that both sides of the equation of the lemma are multilinear (over ((R")) as functions of the vector fields L' and L2. This is true for the left side since the wedge product is multilinear. For the right side, this follows from the computation PROOF

[aiL',a2L2] =

I!

— a2(L2{ai}) L'

+aia2[L',L2], for aI,a2

((RN).

Since both sides of the equation of the lemma are multilinear (over ((RN)) as functions of L' and L2, then it suffices to verify the lemma when I) = 0/Ox3 and L2 = OlOsk for 1 j, k N. In this case, [L', L2] 0 and the equation follows from an easy computation. I

15

Contractions

Contractions

1.5

The contraction operator

i

is

defined as the dual to wedge product.

DEFINITION 1 Suppose L is a vector in The (r — 1)-form L4 is defined by

(RN)

is an r-form at

p E RN.

A...ALr)p.

A...ALr_i)p =

L4 is called the contraction of 0

and

by L.

For example, we have

a .—j(dxj A... x3

(0 =

A... Adxik

ifj=ik.

The notation indicates that dxtk is removed. The contraction operator also satisfies a product rule. LEMMA 1

Suppose L is a vector in T (RN) and suppose belongs to A8T;(RN).

belongs to ArT (RN) and

Then

This lemma is established by first proving it for basis elements, i.e., L = = r and = dxi, 0/Ox,, q51 = dx1, = s. The general case then follows since both sides of the equation are multilinear as functions of L, and The

contraction operator makes it easy to compute the £2-adjoint of the

exterior derivative. The Euclidean inner product (.) on RN naturally extends to ArT*RN by declaring that the collection {dx'; I increasing} is an orthononnal basis. Note that Ox

III=IJI=r,

the pairing on the right is the pairing between r-forms and r-vectors discussed in Section 1.3. Let L be a vector and let be a 1-form that is dual = 1 and iijL' = 0 for all vectors L' which are orthogonal to L). to L (i.e., From the definition of contraction, we have where

for all 0

E

ArT*(RN) and

E

Analysis on Euclidean Space

16

We define an £2-inner product on 7Y(RN), by setting

=

J

1Y(RN).

for

XERN

Dr_I(RN)) is defined by

The £2-adjoint of d (denoted de: 7Y(IRN)

DI(RN).

= LEMMA 2

For

= gdx'

Dt.(1ZN)

dq5= To establish the lemma, we take the inner product of dçb with an arbiFrom the definition of d and d and integration trary = by parts, we have PROOF

= =

(dx' (dxi A

g(x)

J

RN

= —

f

(dxi A

= (dx',

Since we can write

= —

= ((O/0x3)jdx',

h(x)dr')

J RN .7—

IN = I _LJdX 0

I

\

as

desired.

I

j=1

Ox, Ox3

£2

dx

2 Analysis on Manifolds

After one is comfortable with analysis on Euclidean space, the next step is to study analysis on spaces that only locally look like Euclidean space. This leads us to the concept of a manifold. After some definitions, we generalize the notions of vectors and forms from Chapter 1 to the manifold setting. Integration

of forms on orientable manifolds is discussed and the chapter ends with the proofs of Stokes' theorem and some of its corollaries.

2.1

Manifolds

A real N-dimensional smooth manifold is a Hausdorff topological space X together with an open covering { } of X (where runs over some index set) and open maps xQ: Ua —' RN with the following properties: DEFINITION I

(i)

(ii)

is a homeomorphism with its image in RN. fl

Whenever UQ

the

fl

CR".

A smooth manifold locally "looks like" IRN because about any point there is RN an open set (often called a coordinate patch) and a map xü: tic, (often called a coordinate chart) that takes Uc, homeomorphically to an open subset of RN. As we shall see, familiar concepts from analysis on Euclidean space such as differentiation, vectors, and forms can be defined on a manifold via these coordinate charts. In requirement (ii) of the definition, it makes sense to talk about the differo entiability of because this map is defined on an open subset of RN (namely This requirement also distinguishes different categories o of manifolds. For example, if we require to be a real analytic difwe If, instead of feomorphism, then X becomes a real analytic

'7

Analysis on ManifoLds

18

o have C't and we require to be a biholomorphism (i.e., holomorphic with a holomorphic inverse), then by definition, X is a complex manifold. Note that any open subset of a manifold is again a manifold. The easiest example of a smooth manifold is Euclidean space itself. Here, there is only one coordinate patch (i.e., all of RN) and there is only one coordinate chart (the identity map). In the same way, C't is a complex manifold. which is the The next easiest example of a manifold is projective space under the equivalence relation collection of equivalence classes in

if there exists a nonzero real number A with

(xo,...,xN)=A(yo,...,yN). The equivalence class of (xo,... , XN) is denoted [xo,. . , XN} and it is the line in that passes through the origin and the point (xo,. . XN). The coordinate patches are .

. ,

O
(x0

X*([XO,...,XN])= I

—+

RN given by

XN X,

S1

Complex projective space is defined analogously by replacing by and by letting A be a nonzero complex number in the above equivalence relation. More examples of manifolds will follow the definition of a submanifold. The concept of a differentiable function on Euclidean space generalizes to the manifold setting. Suppose X is a smooth manifold. A function f: X

C is

said to be of class Ck (k 0) on X if for each coordinate patch I: {U0 } —p C is of class Ck in the Euclidean sense. function f o

the

DEFINITION 2

The point is that 1 o

x;' is defined on an open subset of Ri" and therefore it

makes sense to talk about the differentiability of I OX;'in the usual Euclidean sense. Requirement (ii) in the definition of a manifold ensures that if f o fl for any other coordinate is differentiable then so is f o °' RN. = (Jo X;') o ° and This follows from Jo chart ° noting that is smooth. The above definition can be generalized to define the concept of a smooth map between two manifolds. DEFINITION 3 Suppose X is a smooth manifold with coordinate charts RN and suppose Y is a smooth manifold with coordinate charts

Submanifoids

19

RN'. Wesaythatf: X

Y isa CIC map if for every 8, is a Ck map from C R" to is c RN', provided 010 Y is C' and if I defined. The map f is called a diffeomorphism if f: X has a C' inverse f_I: Y X. —'

These definitions can be modified for other categories of manifolds. For example, if X is a real analytic manifold, then f: X —* C is called real analytic if for each coordinate chart 10 x;' is real analytic in the Euclidean sense. If X is a complex manifold, then f: X C is called holomorphic if for each c coordinate chart is holomorphic in the Euclidean sense. It is important to realize that different manifold structures can be put on the same topological space by changing the coordinate charts. For example, C'2 is a complex manifold whose usual manifold structure is given by taking the identity map for its coordinate chart. However, suppose we are given a smooth map to R2t2) C'2 C'2 whose real derivative (as a map from x = (x',. . . , at each point in some open set C C'2 is nonsingular. We can define a new complex manifold structure on by declaring x to be the coordinate chart. If x is not holomorphic in the usual sense, then this new complex manifold structure on is different than the standard Euclidean complex manifold structure. A function f: C is holomorphic with respect to this new structure provided is holomorphic in the usual sense, or equivalently, if there is a locally f c' defined holomorphic function F in the usual sense with

f

f=

FOX.

This idea will be used in Section 9.1, where we shall use a nonstandard complex manifold structure for C'2.

2.2

Submanifolds

Next, we define the concept of a submanifold of another manifold.

Suppose X is a smooth N-dimensional real subset of X. We say that M is a smooth (imbedded) of X of real dimension £ if for each point E M, there is a neighborhood U of p0 R" such that in X and a smooth map U DEFINITION 1 a

(i) (U)

(lii)

X: U —, X(U) C

is a

X(Po) is the origin in

X{U fl M} is an open subset of the origin in the copy of Rt given by {(ti,...,tt,O,...,O) E RN;(t,,...,tt) E Rt}.

Analysis on Manifolds

20

The collection of all such xIMr-lu: M fl U —' serves as a collection of coordinate charts for M and thus M is a smooth i-dimensional manifold in its own right. Our main concern will be with submanifolds of Euclidean space. In this case, , xN) is defined on an open subset U of RN. From (iii), the the map = . . . . . . , set MflU can be viewed as the common zero set of the functions , Since x is a diffeomorphism, the derivative has maximal rank at each point in U. This is equivalent to the requirement on

U.

A collection of functions {xt+1,. , with this property will be referred to as a local defining system for M. So about each point of a submanifold, there is a local defining system. The converse also holds. . .

LEMMA I

Suppose M C RN is a subset with the property that for each point p0 in M there is a neighborhood U in RN and smooth functions P1,. . , U —' R so that MflU = {x U;pi(x) = ... = pN_t(x) = 0) and with dp1A. . .AdPN_€ 0 at p0. Then there is a diffeomorphism x = (x',. .. xN) defined near p0, with .

,

= PN-1.

=

In particular,

M is an i-dimensional smooth

submantfold of RN. PROOF

dpN_f

yE

Suppose p0 E M and Pi,... , PN-t are given. Since dp1 A ... A 0 at p0, we can choose coordinates (x,y) for with x E Rt and so that atpo is anonsingular (N—t)x (N—i)

matrix. Then we define x: Ri" _, RN by

x(x,y) = (x,p(x,y)) for (x,y)

Rt x RN_t

where p = (P1,... , pN—t). By the choice of coordinates,

at p0 is non-

singular and so x is a local diffeomorphism by the inverse function theorem. Therefore, has all the properties required for the lemma. I The above lemma allows us to construct many examples of submanifolds of For example, the unit sphere {x RN; — 1 = 0) is an (N — 1)-

dimensional submanifold or hypersurface in RN. Another important example is

the graph of a smooth function. Give RN the coordinates (x, y) with x E Rt RN_t be a smooth function. Define and y RN_Z. Let h: Rt

M = {(x,y)

E

R";y = h(x)}.

Miscalledthegraphofh. where ...

M

is

we have written h = (h1,. .. , hN_t). M is the common zero set of PN—i. Moreover, dp1 A ... A dpN_t is nonzero (everywhere) and so an i-dimensional submanifold by the lemma. In this case, x is given by

21

Submanifolds

x(x, y) = (x, y — h(s)). The converse also holds locally. That is, a submanifold can always be written locally as the graph of a smooth function. LEMMA 2

and p0 is a point on Suppose M is a smooth i-dimensional submantfold of , and a neighborhood U of the M. There is an affine linear map L: origin and a smooth function h: R1 —+ RN_I such that (i)

L(p0) is the origin.

(ii)

L{M}flU—{(s,y)ER1

(iii)

h(0) =

PROOF

0

and Dh(O)

0.

First, we translate coordinates so that p0 is the origin. Give RN the

coordinates (x,y) with x ER1 and yE RN_I. We let p',... ,PN—I be a local defining system for M near the origin. The gradient vectors Vp1,... , are linearly independent at the origin. Therefore, we can find a nonsingular linear N — £. The which sends Vp3 (0) to O/0y2 for 1 map L: RN , manifold L{M} has a local defining system p3 o . , I3N...I} with 13j = and

j

L'

1/,

is the identity matrix. By the with = RN_I with implicit function theorem, there is a smooth function h: Rt h(0) = 0 and

The matrix

p(x,h(x)) =0, for x near 0. Near the origin, L{M} is the graph of h. By differentiating this equation and then using (0) = O/0y3 for 1 j < N — £, we obtain Dh(0) = 0, as desired. I The previous two lemmas describe the way submanifolds are most often presented when doing local analysis. When a submanifold is presented by a local defining system, the functions in that defining system locally generate all functions that vanish on M, as the following lemma shows.

LEMMA 3

Suppose M is an i-dimensional smooth = PN—t = 0} with dp1 A...AdPN_1

of RN given by M =

= R M. Suppose f: is a smooth function that vanishes on M. Then there are smooth functions defined near M so that

I

near

M.

Analysis on Manifolds

22

By a partition of unity argument, we need only to establish the existence of the ü 's in a neighborhood of each point in M. Give RN the coordinates PROOF

(x,y) with x E Rt, y E RN_t. Near a given point in M, we use Lemma I to find a local diffeomorphism x that takes a neighborhood of M to the set {y = O}. Our given function f satisfies f = 0 on {y O}. We shall show

= for some choice of smooth functions cd,. . . The lemma will then follow by composing both sides with and noting that y3 0 X = p3 by Lemma 1. Let we write f = fo X'. To find the

J(x,y) = f(x,y)

—

f(x,0)

f(x,yi,...,yj_i,O...O)

= d

j=I

= The lemma now follows by letting

right.

Yj

be the expression in the brackets on the

I

The concept of a submanifold can be defined for other categories of functions.

Thus, we have the concept of a real analytic submanifold of a real analytic manifold and the concept of a complex submanifold of a complex manifold. The above three lemmas also locally hold in the real analytic and complex analytic categories. We leave the minor modifications of the above proofs to the reader.

We now discuss the concept of a submanifold with boundary. From Definition 1, an £-dimensional submanifold is a set that can be locally straightened to an open subset of Ri. Roughly speaking, an e-dimensional submanifold with boundary is a set that can be locally straightened to a half space in Rt. More precisely, we have the following definition. DEFINITION 2 A subset S of an N-dimensional manifold X is called an dimensional submantfold with boundary if for each point p E S. there is a neighborhood U of p in X and a diffeomorphism x: U with C

23

Vectors on manifolds

=

0 such that one of the following holds:

is an open neighborhood of the origin in {t = (tl,...,tN) RN; tt+i = = tN = 0}. X{U fl S} is an open neighborhood of the origin in {t = (t1,.. . , RN; 0 and = tj.j = 0}.

(i)

(ii)

E E

Both conditions cannot hold simultaneously for a given point p. A point in S that satisfies condition (i) is called a manifold point. A point in S that satisfies condition (ii) is called a boundary point. The set of boundary points in S is denoted OS. Note that OS is an (€ — 1)-dimensional smooth submanifold of X. R is smooth For example, suppose S = {x E RN , p(x) O} where p: RN with dp 0 on S. From Lemma 1, S is an N-dimensional submanifold with boundary in RN. Its boundary is the set OS = {x RN;p(x) = 0}.

Vectors on manifolds

2.3

We now define the concept of vectors in the manifold setting. Suppose X is a smooth N-dimensional manifold and let p0 be a point on X. Let x: U —p R" be a coordinate chart defined on the coordinate patch U containing Here and for the rest of this chapter, we change our notation a bit. We denote the component functions for our coordinate chart x by (x1,... ,XN) instead of XN. We want to think of x1, ... , XN as coordinates on X, rather than just component functions. To distinguish these coordinates from the coordinates in RN, we by denote the latter by t = (t1,. .. tN). For p U, define the vector the equation ,

(IT) p {f}

forf€E(U). )

is smoothly defined on the open subset X { U } of RN, the Euclidean (t) = tk for (tI,...,tN) RN, we have

Since

derivative on the right is well defined. Note that since xk °

t=

ox,

1f

ixkj —

1

0

ifj=k otherwise.

A vector at a point p X is any operator of the form

a3 (i—)

where each

a3 E R.

24

Analysis on

The Set of all vectors on X at p is called the (real) tangent space of X at p and is denoted So each vector in is a linear functional on the space £(U). It is particularly important to be able to identify the tangent space of a submanifold. We have the following lemma. LEMMA

I of RN defined by M = {t E = 0) with dpi A...AdPN_e 0 on M. Fora

Suppose M is an L-dimensional

=

=

point p E Al

L{p,}=Oatpfor 1 j N—L}. PROOF

This lemma has an intuitive geometric proof. From advanced calculus,

each Vp3 is orthogonal to Al. consists of all vectors in RN that are orthogonal to {Vp3(p); 1 j N — L}. Since L{p3} = Vp3 . L (where (.) is the Euclidean inner product), the lemma follows. This lemma also follows from Lemma 1 in Section 2.2. Near p E M we can find a local diffeomorphism = (x1,. . ,xN) with = = . . . PN—e. As mentioned in Section 2.2, the map t (t),. . . , x1(t)) e Re serves as a coordinate chart for M. By definition, is spanned .

Since (ô/ôx3)(xk) = 0 for j k, clearly (O/Ox,)(pk) =Ofor 1 <j
by

us return to the general case of an N-dimensional manifold X. The for p E X is called the tangent bundle of X and is denoted T(X). If X is a complex manifold, then we will refer to T(X) as Let

collection of all

the real tangent bundle in order to distinguish it from the complexified tangent bundle that is defined in Chapter 3. A smooth section of the tangent bundle of X is called a vector field on X. More precisely, a vector field L on X is a smooth function that assigns to each point p E X a vector Here, smooth means that in any coordinate E patch U C X with coordinate chart x: U Rh', L can be expressed as

Lp>ai(P)(/_) x3

for pEU

j=I

where each a,: U —p R is a smooth function. Note that if f is a smooth function

on X and L is a vector field on X, then L{f} is a smooth function on X. We call L C T(M) an rn-dimensional subbundle of T(X) if L assigns to each point p E X an rn-dimensional vector space IL,, C These vector spaces are required to fit together smoothly in the sense that near each fixed point po E X, there are smooth vector fields L1,... , Lm so that (Lm)p} forms a basis for As an example, let X be an open subset of RN and suppose

Vectors on manifolds

25

pl,...,pN_t: X

Rare smooth functions with dp1 A...AdPN_t Oon X. with We let IL,, be the set of vectors = 0 for 1 i N — £. We

leave it to the reader to show that L = UPEXL,, is a subbundle of T(X). By Lemma 1, each is the tangent space at p for the submanifold given by the unique level set of p = (p1,... pN—t) that passes through p. ,

The tangent bundle is an example of a vector bundle over a smooth manifold. We shall not need the definition of a vector bundle in its most abstract form. Instead, we refer the reader to [W2]. Often, different coordinate patches are used near the same point po E X and so it is useful to know how the description of a given vector in (X) changes RN be another as the coordinate chart changes. Let Y = (yi, ... ,yN): U' coordinate chart. For p E U fl U', we describe the vector in terms E

of 0/Oxi,..., O/OXN and O/OYi,.

.

O/OYN

=

Since

(0/Ox,)

we have

b, = = a relation between the a's and the b's. Let L = O/Oxk (so ak = and all other a3 = 0); we have

= and so

fO\

=

1

(O\

N(O\ u—)

{y,}

(O\

We now describe the push forward of a vector under a smooth map between manifolds. This is given in Chapter 1 in the Euclidean setting and the definitions easily generalize to the manifold setting. Suppose F: X —+ Y is a smooth map

between the smooth manifolds X and Y. This induces the push forward map —. TF(p)(Y), which is defined by

=

o

F}, for

E

Analysis on Manifolds

26

Here, g is any smooth function defined in a neighborhood of F(p). If L is a vector field on X then the above equation reads

=

0

F}.

for each p E X. Note that (FOG). = F, oG. for two smooth maps F and C. If F has a smooth inverse, then From the definition of 8/8x3 given at the beginning of this section, 8/ôx,

is the push forward of 0/Ot) (from RN) via the coordinate chart

i.e.,

In Chapter 1, we gave a formula for the push forward of a vector in terms of the derivative of the map. The analogous formula holds for maps between manifolds. Let F: X Y be a smooth map. Suppose p is a point in X. Let RN be a coordinate chart for X with p E U and let U x = (x1,. . .

beacoordinatechartforYwithF(p)E V. Here Y=(yI,...,yN'): N' = dimR Y. We let F) = o F: U R, 1 <j
[F.

OXk

F(p)

which in view of the definition of push forward equals

So

1 k N.

=

If F is the identity map from X to itself, then we recover the change of coordinate formula given above which relates the (8/Ox)-vectors to the (O/Oy)vectors.

2.4

Forms on manifolds

As with Euclidean space, the space of I-forms at a point p E X, denoted T(X), The pairing between forms and is the space of all linear functionals on

Foracoordinatechartx = (xI,...,xN): U RN, we let dx1,. .. , dxN be the dual basis to 0/Oxi,.. . The space of r,

forms at p is the rth exterior power space of all linear functionals on

This can be considered as the A basis

Forms on manifolds

27

(over IR) for A

A

dxsr; I = (ii,. ,ir) is an increasing index of length r}. . .

By definition, the bundle of r forms is called a differential form of degree r, or more

simply, a smooth r-form on X. By smooth, we mean that in each coordinate patch U C X with coordinate chart x := (x1,.. . , xiv): U —* RN, we can express the form as

= III=r

is a smooth function on U. The space of smooth r-forms on X will be denoted By pairing each side with = 0/ox, A ... A we can express the coefficients as where each

OXj

the coordinate description of an r-form depends on the coordinate chart used, as one would expect. Y between manifolds induces the pull back operator A smooth map F: X So

Ft:

This is the dual of the operation of push

—,

forward of vectors, i.e.,

Le For a coordinate chart x = (x1,... , XN), we have = xtdt,, I <j N, where t = (t1,.. . , tN) are the coordinates for IRN. For higher degree forms we define

Ft(01 A ...

Note that (FoG)t

A Or)

= (Ft01) A ...

A

(FtOr),

Øt

E Tt(Y).

Gt o Ft. In particular, if F is invertible then (F_I)* =

Er+I(X) The defiNow we define the exterior derivative d: E?(X) nition given for the exterior derivative of an r-form on Euclidean space is a candidate for the definition in the manifold setting using a coordinate chart

x=

(x1,. .. , XN). However, we must show that the resulting coordinate definition is independent of the coordinate chart. The key lemma is the following. LEMMA 1

Suppose x: U o

RN and Y: V

are

xt{dRN (x0)} = Yt{dRN

two coordinate charts for X. For

on Un V.

Analysis on Manifolds

28

PROOF

The notation daN refers to the Euclidean exterior derivative. The

assertion of the lemma is equivalent to

=

Recall, y1

o Y1 is a smooth map between open subsets of RN. Lemma 2 in Section 1.4 states that the exterior derivative on RN commutes with pull back. Therefore

=daN{(xoY')x'cb} = daN {(x' ox = as desired.

daN

I

The above lemma allows us to define the exterior derivative unambiguously.

The exterior derivative on X dx: fined in a coordinate patch U C Rtv by DEFINITION 1

—'

is de-

dxçt = x*(dRN where x: U

RN is the corresponding coordinate chart.

If the manifold X is clear from the context, then the "X" often will be omitted from the notation of the exterior derivative.

= . XN) is a coordinate chart then (dt,) = dx, for 1 <j
If x = (x1,

. . ,

=

and

(p)dx, A dx' j=1

where we have written

The fact that the exterior derivative on Eucidean space commutes with pull backs generalizes to manifolds. LEMMA 2

Suppose F: X —' V is a smooth map between manifolds. Then dx o F* =

F* ody.

Let x: U —i RN and Y: V —' RN' be coordinate charts for X and V respectively with F{U} fl V 0. Using the definition of the exterior derivative, the statement of the lemma is equivalent to PROOF

x* odRN

o x'

o

F = F*

)J* o daN' 0

Integration on manifolds

29

which in turn is equivalent to dRN o

(F °

x'Y = (Yo F °

o dEN'

oy.

As in the proof of the previous lemma, this equation follows by commuting the exterior derivative on Euclidean space with the pull back of the map Y o F o

I

RN

As an example, suppose X = {x E RN; p(x) = O} where p: R is smooth with dp 0 on M. Let j: X —' RN be the inclusion map. Then 0 dRN = dx In particular, j*dRN (p) = 0 because jp = p o j = 0 on X.

2.5

Integration on manifolds

We first discuss orientation. The form dt1 A.. .AdtN on RN induces the standard orientation on Ri". A basis of 1-forms . . ØN for T (RN) is said to be oriented if A.. = A.. .AdtN with > 0. A linear map L: RN _, RN is said to be orientation preserving if . LØN is oriented whenever ØN is oriented. This is equivalent to the condition that the determinant of the matrix that represents L is positive. A smooth map F: RN ., Ri" is said to be orientation preserving if F* (p): i'; (RN) is orientation preserving for each point p in its domain of This is equivalent to the condition that dat DF(p) > 0 for each point p E RN in the domain of definition .

. ,

ofF. Let X be a smooth N-dimensional manifold. A collection C of coordinate o U —' RN} for X is said to be orientation preserving if is orientation preserving as a map from Ri" to Ri" for each 3) belonging charts

to C. The manifold X is said to be orientable if there is a collection of orientation-preserving coordinate charts such that the corresponding coordinate

patches cover X. If X is a submanifold with boundary, then we define X to be onentable in the same way. It is an easy exercise to show that if a manifold with boundary is orientable then its boundary is also orientable. Every complex manifold is orientable. For if F is a holomorphic map be-

then detDF = Idet(OF/Oz)j2. Here, DF tween any two open sets in is the real derivative of F as a map from whereas OF/Oz is the to complex derivative of F as a map from to C'2. Therefore, roy' is orientation preserving for any holomorphic coordinate charts x and 3) on a complex manifold. If the manifold X is orientable, then its collection C of orientation-preserving coordinate charts determines an orientation for T (X) as follows: a set of one A ... A ØN at p is a forms .. } E T; (X) is said to be oriented if ,

AnalysL on Manifolds

30

positivemultipleofdxlA...AdxN belongs to C. Since o is

T;(X) for each p

= (xI,...,XN): U _.RN Y x E C. With such an orientation for

X, we call X an oriented manifold and we call the chart

x E C an oriented coordinate chart.

If M is an (-dimensional oriented submanifold with boundary contained in an N-dimensional manifold X, then we define the induced orientation on RN be an oriented boundary For p OM, let x = (x1,. . , xx): U C X coordinate chart, i.e., a coordinate chart x C with = 0 and = {(t1,. . = O}. Let n = —dxj = E RN;ti 0 and ti-f I = = tN This 1-form is called the outward pointing co-normal for OM. If (with p E V) is another oriented boundary coordinate 3': V c X —' chart on M, then at p is a positive multiple of dx1 at p. Hence, we can make the following unambiguous definition: a collection of one forms c T;(OM) is said to have the induced orientation on provided {n, has the orientation on M, i.e., if nAØ1 A. . is , .

.

. .

a positive multiple of dx1 A. . .Adxt. This orientation differs from the orientation

on OM obtained by restricting the oriented coordinate charts for M to OM. A collection of forms has this orientation if A... A is a positive multiple of dx1 A ... A or equivalently, if Øi A... A A n is a positive multiple of dx1 A ... A dx1_1 A n. Since

nAdx1 A...Adx1_1 =(—1)'dxi A...Adxt, induced boundary orientation on OM differs by a factor of from the orientation obtained by restricting the oriented coordinate charts from M to OM. As an example, let M = {(xI,x2,x3);x3 0}. Then OM = {(xi,x2,0)} and n = —dx3. The orientation on OM as a subspace of R3 is determined by dx1 A dx2. However dx1 A dx2 has the opposite orientation from the induced orientation on as the boundary of M. Now we define the integral of a differential form. For a D(RN), we define the

RNRN where the right side is computed in the usual way.

Suppose U and V are open sets in RN and let F: U V be a diffeo(a 0 F)(detDF)dx where dx = morphism. From Section 1.3, F(adx) dx1 A ... A dXN. Using the language of differential forms, the change of variables formula for integration is

Jodx =

{

fF(adx) ifDetDF>0 -fF(adx) if Det DF <0.

Now suppose X is an oriented N-dimensional manifold and let

be a smooth

Integration on manifolds

31

RN

N-form with support contained in a coordinate patch U of X. If x: U is the corresponding oriented coordinate chart, then define

= formally define the integral of an r-form on an N-dimensional manifold to N. be zero if The first thing to check is whether or not the above definition is independent of the oriented coordinate chart. If Y: V is another such chart with supp C V, then we must show We

I Since

)) o

RN

x' is

det D(Y o gration, we have

RN, a smooth, orientation-preserving map from > 0. Using the above change of variables formula for inte..

RN

=J x'YY'ø x'ø RN

desired. We conclude that the integral of a form supported in a coordinate patch is unambiguously defined. Suppose is a compactly supported N-form on X (but not necessarily supported in one coordinate patch). Suppose {U1,... , } is a collection of coRN be the corresponding ordinate patches that cover supp and let U, oriented coordinate charts. Let ... , } be a partition of unity for supp with c5j E V(U,). Then we define as

I

X

=

Again we check that this definition is independent of the choices made. Suppose {Y,: V3 Rh', j = 1,..., n} is another set of oriented coordinate charts whose corresponding coordinate patches cover supp Let . . , } be a

Analysis on

32

partition of unity for supp 0 with

Since

E

t'k

1,

= k=I)=IRN

has support in a single coordinate patch, we have

Since

RN

Summing the right side over j and using

I, we obtain

k=IRN

so our definition of the integral of 0 is well defined. If M is an £-dimensional submanifold of X, then we define

and

10 where

j: M

—'

X is the inclusion map. The point is that

is

an intrinsically

defined form on the £-dimensional manifold M and so the right side can be evaluated as described above. If 0 contains a factor of dp where p: X —' IR vanishes on M, then JM must also vanish since j*dp = dj*p = 0 on M. Also if r $ £, then JM 0 = 0 for 0 77(X). If M is a smooth £-dimensional submanifold with boundary contained in an N-dimensional manifold X, then fM 0 is defined in a similar way to the case when M is a manifold. Here, the only difference is that M has two types of coordinate charts. The first type is a coordinate chart about a manifold point and the second is a coordinate chart about a boundary point. If x: U —' is a coordinate chart about a manifold point, then fl M} is an open subset of = {(tI,...,tN);tt+I = = tN = 0}. If 0 is an €-form with support in U fl M, then by definition,

10 =Jx'(o). If x:

U

—'

RN is a coordinate chart containing a boundary point, then

is an open subset of the half space of IR' given by {(t1, . t,+i by definition,

f0= J tER'

M

ttO

(x'O)(t).

. . ,

tN); t1

0 and

Integration on

33

As with the case of a manifold without boundary, the above definitions are independent of the oriented coordinate chart. If 4> is a compactly supported i-form on M (but not necessarily supported in one coordinate patch), then its integral is defined in the same way as the case where M has no boundary by using a partition of unity. Now suppose X and Y are smooth, oriented N-dimensional manifolds and suppose F: X —' Y is a smooth map. We say that F preserves orientation if preserves orientation on R" for each oriented coordinate chart x Yo F o for X and Y for Y. The change of variables formula for easily generalizes to manifolds.

x'

LEMMA 1

Suppose X and Y are smooth, oriented N-dimensional Suppose F: X Y is a smooth, orientation-preserving map. 114> E D"(Y), then

Jq5=JF*çb. The proof of this lemma follows from the definitions together with the change of variables formula on 1W". Details are left to the reader. We are now ready to state and prove Stokes' theorem, which for an oriented

submanifold M with boundary, equates the integral of dçb over M with the integral of 4> over ÔM. The simplest example of Stokes' theorem is the fundamental theorem of calculus, where M is the interval {a x b} in lit We shall show that the definitions of forms, orientation, and integration reduce the proof of Stokes' theorem for more general manifolds to the fundamental theorem of calculus. THEOREM 1

STOKES

Suppose M is a smooth, oriented e-dinwnsional with boundary that is contained in a smooth N-dimensional manifold X. Suppose 4> then

f

dM4>

Here, ÔM has the induced boundary orientation.

PROOF We first cover supp 4> with a finite number of oriented coordinate C X '—' RN i = 1,... ,m that are either of manifold type (i.e., charts x': xl{Ui fl M} = {(t1,... ,tN);tt+I = = tp,r = 0)) or boundary type (i.e., = {(tj,. ,tN);tt = ... = V(U1)} = 0}). Let . .

be a partition of unity for supp 4>. It suffices to prove JdM{4>i4>}

=1

Analysis on Man(folds

34

for each i = 1,... , m. Stokes' theorem then follows by summing over i and using 1 on supp So from now on, we assume that supp 0 is contained

in a coordinate patch (one of the We first consider the case where x: U fl M —. R1 is a coordinate chart of manifold type. Using the definitions of the integral and the exterior derivative on M, we have

f is a

d refers to the exterior derivative on IRe. Since compactly supported (C— 1)-form on 1R1, we may write

A ... A dt3 A ... A dt1

(to)

belongs to V(Rt). The notation omitted. We have where each

indicates that dt3

has

been

So

...f The last equality follows from the fundamental theorem of calculus in the t3-

has compact support. variable, using the fact that Therefore, we have shown that if has support in a coordinate patch, U, of = 0. On the other hand, we have U fl OM = manifold type, then also vanishes and we have established Stokes' theorem: 0. Therefore, ó in this case. = The remaining case to consider is where o has support in a coordinate patch U of boundary type. In this case, we have

f

J

M

,({UnM}

=

f ttO

Integration on

35

As above, we write

where

E

V(Re). We claim that for each j = 1,...

f

.

f

.Adt1} =

(t,O}

.

(1)

is just the special case of Stokes' theorem where M is the half space {tt O} because, as mentioned earlier, the induced boundary orientation of

This

{tt = 0} differs from its inherited orientation from Rt by the factor (—I )t. If j
f

J

{ttO}

=0. last equation follows by the fundamental theorem of calculus in the t3variable, using the fact that has compact support. Since j <€, there is no boundary term. If j < e, then the right side of (1) also vanishes because the pull back of dt1 to {t1 = 0} is zero. Thus, both sides of (1) vanish for j < e. The

If j = £, then

f

f

A...Adtt_i}

(eeO}

=(_l)t

dt1

f {tt=O}

where the last equality follows from the fundamental theorem of calculus in the

t1-variable. This proves (1) for j = Summing (1) over j yields

f

{te>O}

= (_1)t

f

Therefore, Stokes' theorem follows for the case of a coordinate chart x: U fl Rt of boundary type. The proof of Stokes' theorem is now complete. I M We

the divergence theorem and Green's forbe an open subset of Rv with smooth boundary. Define

prove two useful corollaries:

mula. Let M =

36

Analysis on Mantfolds

p: RN

R by —

(

f —dist(x, RN

jf x E

—

It is an easy exercise using the inverse function theorem to show that p is smoothly defined on a neighborhood of Oft Define the vector field N Vp N is the unit outward normal to The volume form for Oci is given by the (N — 1)-form

cia =

where

dx =

= COROLLARY I

dx1

A ... A dXN

A... A

A... A dXN.

DiVERGENCE THEOREM

is a bounded open set with smooth boundary in RN. Let F = F,(O/Ox,) be a smooth vector field on ft Then

J(F where Div F

N)da = J(Div F)dx

0F3/Ox, and F . N =

is a smooth (N—i)Since N has unit length, Njdp = ion Oft So form on O), then from the product rule for (see Lemma i in Section 1.5), we have PROOF

i

Nj(dp A

=

—

dp

A (N4)

= From the definition of the integral over a submanifold, we obtain

J dpA (Nit)

=

Jf(dp)

=0 because j*dp =

d(jp) =

0 on

Together with the previous equation, we

have

I Applying this equation to the form 5=

.

.

.AdXN

Iiaegration on manifolds

37

yields

... A dx, A ... A dxN

=J(F. On the other hand, by Stokes' theorem

J

A... A dx, A... A dxN j=I

= J(Div F)dx.

The proof of the divergence theorem is now complete.

I

COROLLARY 2 GREEN'S FORMULA

Suppose ci C RN is a bounded domain with smooth boundary and suppose u and t' are smooth functions on ci. Then

J(uNv

—

vNu)do =

—

where N is the unit outward normal to Oci. PROOF We apply the divergence theorem to the vector field

Analysis on Man4folds

38

to obtain

J v(N

Vu)da

=

f

+J

dx.

Next, we apply the divergence theorem to the vector field N

G

u(Vv) =

ox3 Ox3

to obtain

f u(N. Vv)da = J

+f

dx.

Green's formula follows by subtracting these two integral equations.

3 Complexified Vectors and Forms

Objects such as

0

1(0

.0

or

= dx3 + idy, are often encountered in complex analysis. The former is a complexified vector and the latter is a complexified form (due to the presence of i In this chapter, we make these notions precise and generalize some of the concepts in Chapters 1 and 2 to the complexified setting.

3.1

Compiexification of a real vector space

Suppose V is a real vector space. The complexification of V is the tensor product V®R C (or for brevity, V®C). As a vector space over the reals,

isgeneratedbyv®l andv®iforvE V. vector space by defining V.

C is generated over C by v ® 1 for v E V. If V is a vector space of real dimension N, then dimR V ® C = 2N and As a complex vector space, V

dimc V ® C = N. For shorthand, we write v

v v v ® 1. In this way, V is naturally imbedded into V® C by identifying

V with V ® 1. There is also a natural conjugation operator for V 0 C. This is defined by

39

40

Complexified Vectors and Forms

As an example, let M be a smooth manifold of real dimension N. For p E M, C is called the complexified tangent space and T(M) ® C is called the complexified cotangent space. The space i'; (M) ® C can be viewed as the complex dual space of C by defining the pairing ®

L®

= (0, E C.

L E

If F: M —p N is a smooth map between smooth manifolds, then the push forward and pull back operators extend to the complexified setting in a complex = F*q5øcx linear fashion. We define and = for L E and C. Note that these operators commute 0E with the conjugation operator. The complexified tangent bundle Tc (M) is defined analogously to the real tangent bundle. We let Tc (M) = UPEMTP(M)®C. Similarly, the complexified cotangent bundle is defined by T*C (M) UPEMT (M) ® C. A complexified vector field L on M is a smooth section of Tc (M). This means that L assigns belonging to øC. In any smooth coordinate to each p e M a vector we can express L by : Uc = (XI,...,XN) system x

= where each a3 is a smooth, complex-valued function defined on U. A subbundle

L of Tc (M) of complex dimension m assigns a subspace IL,, of 0C of complex dimension m to each point p in M. These subspaces are required to fit together smoothly in the sense that near each point p0 E M there are m linearly independent smooth complex vector fields L',. . , such that is .. , generated (over C) by If M is a complex manifold of complex dimension n, then it is important to .

distinguish between the real tangent bundle and the complexified tangent bundle.

The former, denoted T(M), is defined in Chapter 2, where we think of M as a smooth manifold of real dimension 2n. Its fiber, has real dimension 2n. The fiber of the complexified tangent bundle is 0 C and this has complex dimension 2n. The rth exterior power of

T(M) 0 C are called the spaces of complexified r-vectors and r-forms at p. respectively. The latter can be viewed as the space of complex linear functionals on the former by using the same pairing formula given in Section 2. Letting the point p E M vary, we obtain the bundle of r-vectors ArTC (M) and the bundle of r-forms ATT*C (M), respectively. The space of smooth sections of ArT*C (M) is called the space of complex differential forms of degree r (or, more simply, the space of smooth r-forms) and is denoted by e"(M). An element of eT(M) can be expressed

Complex structures

41

(x1.. XN) : U C M

in local coordinates x =

by

Ojdx'

0= III=r

where each çbj is a smooth, complex-valued function on U.

The exterior derivative and the integral easily generalize to the complex setting. In local coordinates, we define dOIAdxIEEr+I(M)

dq5= =

where

dØj = The proof that the exterior derivative is independent of the choice of local coordinates is unchanged from the proof for the real case given in the previous chapter. The integral of a complex differential form over a manifold is defined

the same way as in the real case. Of course, the result of such an integral is generally a complex number rather than a real number.

3.2

Complex structures

DEFINITION 1 Suppose V is a real vector space. A linear map J: V V is called a complex structure map if J o J = — I where I: V —' V is the identity map.

A complex structure map can only be defined on an even-dimensional real vector space, because (detJ)2 = (_i)N where N = dimR V. As an example, let V = We give the coordinates (xi,yi,... ,Xn,yn). The standard complex structure for is defined by setting

0

'\oxj)

0y3

\0y3J and then by extending J to all of

Ox3

by real linearity. This complex structure map is designed to simulate multiplication by i = The standard complex structure map for is an isometry with respect to the Euclidean metric (.) on as the following lemma shows.

Complexified Vectors and Forms

42

LEMMA 1

w = —v• Jw. In particular Jv Jw = V W.

For v,w E

This lemma is established by showing that it holds when v and w are basis vectors (i.e., 0/Ox3 or O/Oy,). is defined The standard complex structure J* for the cotangent space T we have as the dual of J. For p (J*Ø,L)P = Setting

for

E

and L e

= dx3 or dy, and L = 0/Ox, or O/0y3, we obtain J*dx, = —dy,

A complex structure can be defined on the real tangent bundle of a complex up to M via a manifold M by pushing forward the complex structure from coordinate chart. For p E M and holomorphic coordinate chart Z : U C M —' Tp(M) (with p E U), we define the complex structure map .J, :

by

= It folHere, the J on the right is the complex structure map on with z3 = x3 + iy,, then lows from this definition that if Z = (z1,... , is = —(0/Ox3). This description of = O/Oy, and is independent of the choice of holomorphic coordinates. For if W: U' another holomorphic coordinate chart, then Z oW' is a holomorphic map from to itself and therefore its push forward map commutes with J (the complex —' T,,(M) is well defined for each p E M. structure on Ca). Thus, can be defined on T (M). In a similar manner, a complex structure We now return to the general case. If J V —+ V is a complex structure for the real vector space V, then J can be extended as a complex linear map on the complexification of V by selling

J(av) = aJ(v) for v

V and a E C.

Note that for

v€V®C

for if v1 e V and a E C, then

=

by the definition of conjugation

= ãJ(v1) by the definition of J on V 0 C = aJ(v1)

= J(avi).

Complex structures

Since

43

JoJ

—Ion V. the same holds true on V®C. Therefore, J

V 0 C —' V 0 C has eigenvalues +i and —i with corresponding eigenspaces denoted by V"° and V°". We have

V0C= from elementary linear algebra. Since Ji) = Jv for v E V 0 C, we have

v°". V

A basis for V"° and for V°" can be easily constructed. First note that for V. the vectors v and Jv are linearly independent over IR because J has no

E

real eigenvalues. We can inductively build a basis for V (over IR) of the form

v1,JvI,...,vn,Jvn where 2n = dimR V. We claim the set

is a basis for the complex-n-dimensional vector space V"°. Each vector v3 — iJv, belongs to V"° because J(v3 — iJv3)

=

—iJ2v3

= i(v,

—

+ Jvj

iJv3).

Moreover, this set of vectors is linearly independent over C. Likewise, the set

is a basis for V°'. We return to the example where M is an n-dimensional complex manifold. Let (z1,... , with z3 = x3 + iy3 be a set of local holomorphic coordinates for M. As mentioned earlier, the complex structure map for satisfies = O/Oy, and = —(0/Ox)). Define the vector fields

i/O .O\ and —= 0 8 i/O .O\ l<3
OYj j

In view of the above discussion, a basis for and a basis for is given by In a similar manner, define the forms

A basis for given by

is given by {dzj,... ,

is given by {O/Ozi,. .

.

,

. .

,

and a basis for T°'1 (M) is

}

Complexified Vectors and Forms

44

The constants are arranged so that

0\

/

Ii ifj=k

I

0

I

-

0

\

/

-

8

Ii ifj=k

\

0 0 for

for

E

and

E

LE

=

and

0

T;°'(M) and X E

The Hermitian inner product (.) on

10

0

0 C is defined by declaring that

0

0

is an orthonormal basis. By the definition of a Hermitian inner product, we have

(aU).V=a(U.V) and

Cand U, V E

are

orthogonal under this metric. We can identify

with

via the map

0

for U =

C's. With this identification, we have U. V =

We have also used (.) to denote the Euclidean inner product on IRN. It will be clear from the context which inner product (.) refers to. On the (rare) occasion in which both inner products are discussed simultaneously, we will denote the by H(U, V) and the Euclidean inner Hermitian inner product of U and V E product of U and V by S(U, V), where in the latter case, we view U and V as vectors in We leave the proof of the following lemma to the reader. LEMMA 2

For U, V E

H(U,V) = S(U,V)+iS(U,JV).

In an analogous fashion, the Hermitian inner product on T denoted by (.)) is defined by declaring that

{dzi,. is an orthonormal basis.

.

0 C (also

Higher degree complexifled forms

45

Let us return to the case of a general even-dimensional real vector space V. The construction of V1'° and V°" from a given complex structure map can be reversed as the next lemma shows. LEMMA 3

Suppose V is an even-dimensional real vector space V. Suppose L is a complex subspace of V ® C with the following properties (i)

LflL={O}

(li) LeL=V®C. Then there is a unique complex structure map J on V so that L and E are the +i and —i eigenspaces of the extension of J to V 0 C. PROOF We first define JC : V ® C -. V 0 C by

J'(L)=iL for LEL J'(L)=—iL for LEE Property (i) for L shows that JC is well defined. Property (ii) for L shows that JC can be extended to a complex linear map defined on all of V®C. Clearly, L arid L are the +i and —i eigenspaces for JC• It remains to show that jC maps V(= V® 1) to itself, for then we can define our desired J to be the restriction

of J' to V. Note that the vectors

X=L+L Y—i(L—L)

LeL LEL

are real, i.e.,X=XandY=Y. Hence,XandYbelong to V®!. Since L e E = V ® C, V is generated by vectors of the form X and Y given above. = —X, which both belong to V. clearly jC maps Since J'X = Y and V to itself, as desired.

I

Sometimes in the literature, attention is focused on the —i eigenspace of J and therefore it is denoted by L We have chosen to denote the —i eigenspace by L since it seems more natural to think of vectors of type (0, 1) as conjugated rather

than unconjugated. This is consistent with the fact that the set {O/0, 1 <

j

< n} is a basis for

3.3

Higher degree complexified forms

r 2n, we have Let Al be a complex manifold of dimension n. For 0 defined ATT (M) to be the bundle of complexified r-forms and e'(M) to

Vectors and Forms

46

be the space of r-forms on M whose coefficients are smooth, complex-valued functions. For 0 p, q

n and a point z

M, define the space

=

(M)}.

denotes the antisymmetric tensor product. Let (z1,... , local holomorphic coordinates for M. In more sensible terms, the vector space spanned over C by the set Here,

{dz' A where

A... A

= dz21

I = {i1,. ,i,} and J = {ji,. . .

A

A...

be a set of is

A

run over the set of all increasing

. .

multiindices of length p and q, respectively. In the same way, we define the space

= which is spanned (over C) by

10

0

M vary, we obtain the bundles AP'Q(T(M)) and in the usual way. The space of smooth sections of is

By letting the point z

denoted £P'Q(M) and is called the space of differential forms on M of bidegree can be expressed in local coordinates as (p, q). An element of

> III=p

IJI=q

where each

is

a smooth, complex-valued function.

In the local coordinates z3 = x3 + iy3, we write dx, = (l/2)(dz, + and dy) = —(i/2)(dz, — di,) for I j n. In this way, any element of (M)) can be written uniquely as a sum of forms of various bidegrees,

i.e., Ar(T*C

e .. . e

(M)) =

Therefore

= ForO

p,q

n with p+ q = r, let Ar(T*C (M))

be the natural projection.

.

•. e

47

Higher degree complexzfied forms

The Cauchy—Riemann operator 0 —i and the operator 0: are defined by DEFINITION 1

=

o d

=

o d.

For a smooth function f : M —i C, we have

df =

-1-dx3

+

which can be rewritten

+

df

= The first term on the right is an element of £"°(M) and the second term is an element of (M). Therefore

Of =

Note that df = Of + Of. Also note that a C' function f : M

C is

holomorphic if and only if Of = 0. For higher degree forms, we have

0{fdz' A

= Of A dz1 A

A

= Of A dz1 A

for

f

so

df = Of + Another important observation is that for This follows from 0= from noting that (00 + OO)çb and

= 0, =

=

0, and

=

—OOçt.

and + (00 + OO)4 + have different bidegrees ((p +2, q),

(p + I, q + 1), and (p, q + 2), respectively). We summarize the above discussion in the following lemma. LEMMA 1

-

(i)

d=0+O

(ii)

02=0,00=—OOandO2=0

The operators 0 and 0 satisfy a product rule whose proof follows from the product rule for the exterior derivative (Lemma 1 in Section 1.4).

Complexified Vectors and Forms

48

LEMMA 2

1ff

EP.Q(M) and g

then

The pull back of a smooth map between smooth manifolds M and N preserves degree (see Section 1.3). If M and N are complex manifolds and the map is holomorphic, then its pull back also preserves bidegree, as the next lemma

shows. LEMMA 3 Suppose M

and N are complex manifolds and F: M —, N is a holomorphic

map. If çb is an element of PROOF

then

is an element of

isasetoflocalcoordinatesforNandifF, =w,oF: F*dwj =

dF3

and

F is holomorphic, OF = 0. Conjugating this gives OF, = the equation d = 0 + 0, we obtain F*dw, = OF3

0.

So from

A1'0T(M)

and

= If

01F,

A°"T(M).

= p, IJI = q, then F*(dwIAthi,J)

Since {dw'Ad&;

=

p' IJI = q} is a local basis for AP'QT(N), the proof of the lemma is now complete. I In Lemma 2 of Section 1.4 we showed that the pull back operator commutes with the exterior derivative. If the map is a holomorphic map between complex manifolds, then the pull back operator also commutes with 0 and 0, as the next lemma shows. LEMMA 4 Suppose M

and N are complex manifolds and F: M oO = 0 o F*. and

N is a holomorphic

map. Then F* 0 0 = 0o

The 0 on the left side of the equation F* 0 0 = Oo F* is the 0-operator for

N and the 0 on the right is the 0-operator on M. Since F* commutes with the exterior derivative and since F* preserves bidegree, the proof of the lemma follows easily.

Higher degree complexifted forms

49

For functions, there is a useful relationship between the exterior derivative,

8, 8, and J' — the complex structure map on the space of 1-forms. LEMMA

S

1ff is a smooth,

complex-valued function defined on a complex manifold M,

then

Of =

—

iJtdf)

Of= The operator J o d is often denoted dc in the literature. Thus, (1 /2)dC = (1/2)J* o d is the imaginary part of the 8 operator. We start with the equation d = 0 + 8. For a function f, the one to T" (M), which is the +i eigenspace of J'. Likewise, Of belongs to T'° (M),_which is the —i eigenspace of J. By applying J to the equation df = Of + Of, we obtain PROOF

form Of

J'df = iOf — i8f. Adding this equation to the equation idf = iOf + i9f and then dividing the result by 2i yields the first assertion of the lemma. The second assertion is derived similarly. I We end this chapter with the computation of the £2-adjoint of 0 on C'2. This is analogous to the computation of the £2-adjoint of the exterior derivative on RN given in Section 1.5. The Hermitian inner product for T*C (C'2) can be extended to an inner product on (C'2) by declaring that the set

{dz' A

Ill

= p, IJI = p, I, J increasing}

is an orthonormal basis for pactly supported elements of £2-inner product

=

J (0(z)

be the space of com-

Let We endow

with the following

E

where dv = dx1 A dy1 A ... A A is the usual volume form for C'2. The adjoint of 0 with respect to this inner product is denoted 0' : (C'2) It is defined by and g€ fE (C'2). The coordinate formula for 0' involves the contraction operator whose definition in Section 1.5 easily generalizes to the complex setting. Note

Complexified Vectors and Forms

50

that

for Ill =

p, IJI = q,

0

(0

-J

if j

E J.

Here, we assume J = {ji,. . . and that if j belongs to J then j = In this case, J' is defined to be the index of length q — given by jk• ,jk,. . ,jq} where ik indicates that jk is removed. We now state the for1

.

mula for 0 whose derivation is similar to the proof of Lemma 2 in Section 1.5 and is left to the reader. LEMMA 6 Let = fdz' A Then

where f is a smooth, compactly supported function on

=

—

3=1

4 The Frobenius Theorem

In this section, we discuss the Frobenius theorem. We also discuss the complex analytic version of this theorem, which will be used in Part H when we discuss imbeddmgs of real analytic CR manifolds.

4.1

The real Frobenius theorem

Let L be an rn-dimensional subbundle of the real tangent bundle to RN. The Frobemus theorem gives conditions on the subbundle L in a neighborhood U of a given point p0 which guarantee the existence of locally defined smooth N — m and R with p3(po) = 0, 1 functions P1,... ,PN—m : U on U such that dp1A ... A dPN_m

j

on U

for

exist and satisfy (1). For each c= (c1, Let us assume Pie... , near the origin, the set

(1)

. . .

,

CN_rn)

E RN_rn

= {x E U C RN;pi(x) = C1,... ,PN_m(X) = CN_m} is an rn-dimensional submanifold of U by Lemma 1 in Section 2.2. By Lemma 1

is generated by those vector in Section 2.3, the tangent bundle of each fields on RN which satisfy condition (1) on An easy argument using the inverse function theorem or Lemma 1 in Section 2.2 shows that the family of submanifolds c E RN_rn } fills out a possibly smaller open set U' in which contains p0. It follows that the existence of pi, . .. , PN—?-SL leads to a foliation of an open set in RN containing po by submanifolds of dimension m whose tangent bundle can be identified with L

51

The Frobenius Theorem

52

If L1

and L2 belong to IL, then L1{p,}

= L2{p,} =

0

on U for 1 <

N — m. Therefore [L1,L2}{p,} =

L1{L2p,} —

L2{Lipj} = 0

on

U.

The vector field [L1, L2] satisfies condition (1) and so [L1, L2] also belongs to L A subbundle L is said to be involutive if [L1, L2] belongs to L whenever L1 and L2 belong to L So the existence of p',... , PN—m satisfying (1) implies that L is involutive. The Frobenius theorem states that the converse holds. For simplicity, we state

this theorem for Euclidean space. However, since it is local in nature, it can easily be generalized to the manifold setting. THEOREM 1

FROBEN1US

Let L be an rn-dimensional subbundle of the real tangent bundle of RN. Suppose L is involutive, i.e., [L1, L2] belongs to L whenever L1 and L2 belong to L Then given a point p0 E RN there is a neighborhood U of p0 and a d(ffeomorphism X=(Xt,...,X1V):U_4 such that

on U for

and LEL.

We assume the given point p0 is the origin. Near the origin, there is set of linearly independent vector fields L',... , a that span the subbundle L over e(RN). Give the coordinates (y, x) with y E Rm and x E PROOF

We write

=

+

'7jk

I

j

m

and are smooth functions defined near the origin. Since are linearly independent near the origin, we may reorder the coordinates if necessary so that the m x m matrix /1 = is nonsingular near the origin. Multiplying through by yields another locally defined basis for

where the

L',.

. . ,

L of the form LI,...,Lrn

V=

where

+

A, k

/-

1

j

m.

Each is a smooth function of (y, x) and equals the (j, k)th entry of the matrix product where 'y is the m x (N — m) matrix with entries 'yjk.

By explicitly computing [Li, Lc}, it is clear that the (O/Oy,)-coefficient of [L', Lk] vanishes. On the other hand, by hypothesis, [V, Lk] is a linear combination of { L1,. . , }. Any nontrivial linear combination of { L',. . , Lm } must involve a nontrivial linear combination of {O/Oyi,. .. , .

.

The real Frobenius theorem

53

Therefore, we conclude [Li, Led] = 0

near 0,

1 j,

m.

k

The proof of the Frobenius theorem will be complete after we prove the following key lemma, which is important in its own right. LEMMA 1

Let 1 m

N and give RN the coordinates (y,x) with y

Rm and x

RN_rn. Suppose N-rn

a

k=i

k

where each A3k is a smooth function defined near the origin. In addition, suppose [L3, = 0 for 1 j, k m. Then there is a neighborhood U of the origin and a smooth diffeomorphism x = (xi,... xiv) : U X{U} C R" such that ,

L'{Xk}=O on

U,

and

x(0,x) = (O,x) for x E RN_rn with (0,x) EU. PROOF

The proof proceeds by induction on m, 1 m

N. We first consider

the case m = 1. Of course, for any vector field L, the condition [L, L] = 0 always holds, and so this condition provides no new information. We shall construct a smooth map x = (xi,..., xN) such that L = O/Oxi where the vector field was defined in Section 2.3 as the push forward of

O/Oti under the map x'. Since (O/Oxi){x,} = 0 for 2 <j N, complete the proof of the lemma for the case m = 1. Let (t,x) be coordinates for RN with t R and x local diffeomorphism F = (F1,... ,FN) : R x

=

this

will

RN_i. We shall find a with

for (t,x) near the origin

(2)

F(t,x)

and

F(0,x) = (0,x). Then the map x = F' will satisfy the initial condition x(0,x) = (0,x). Moreover,

0 as desired.

(O\

The Frobenius Theorem

54

In the coordinates (y, x) = F(t, x), we have

= (see Section 2.3). Comparing this expression with

L=

+

we see that (2) is equivalent to the following initial value problem

aFi(t,x) 0F2(t, x)

OFN(t, x) Ot

—

= )ti,i(F(t,x))

(3)

=

F(O,x)=(O,x) for The existence and uniqueness theorem from ordinary differential equations guarRN which is defined on {ItI < antees a smooth solution F R x RN_i < f} for some sufficiently small > 0. Note that the initial condition €, F(0, x) = (0, x) implies that F(0, 0) = 0 and :

_f 1 ifj=k '

0

Together with the above differential equation at t = 0 and x = A11(O)

1

0,

we obtain

0

DF(0,0)=

0 1

In particular, DF(0, 0) is nonsingular. By the inverse function theorem, F has

a smooth locally defined inverse x : RN _, with x(°) = 0. As mentioned above, x is the desired map for the lemma for the case m = 1. The geometric interpretation of the above analysis is the following. For fixed x, the curve {F(t, x); ti < is called an integral curve for L. The vector field L = is tangent to this curve. The solution to (2) determines a unique family of integral curves for L which satisfy the given initial condition.

The real Frobenius theorem

55

N) and we prove the lemma for m. So by assumption, there is a local diffeomorphism = Now we assume the lemma is true for rn —

(ii,

,

N) : RN

1

(2

rn

RN such that

for

(4)

and

(O,x), for x E RN-rn near the origin. Let F be the In particular, = 0, we have F(0) = 0. From (4), we see that V is inverse of Since form k N. In particular, for 1 <j <m — 1, tangent to any level set of is tangent to the hypersurface

M0 = {F(t',O.x); t' E

XE

= 0).

which is the level set

Let (t', trn, x) be the coordinates for RN where t' = (t1,... ,tm_i) E tm E R, S = (x,,. . ,XN_rn) E RN_rn. We find a local diffeomorphism .

F:Rrn1 XRXRN-m

With

=

[F. rn

(5)

F(t,x)

F(t',O,x) = F(t',O,x) t' E

XE

This is accomplished by solving a system of ordinary differential equations that is analogous to (3). The solution to (5) determines a family of integral curves for that satisfy the given initial condition.

We claim that F is a diffeomorphism on a neighborhood of the origin in F(t',O,x) This follows from two observations. First, the map (t',O,x) pararneterizes M0. Second, the map trn F(t', trn, x) parameterizes an integral curve for Lm that is transverse to Mo (because Lm is transverse to L1, . , . .

the set {(0,x);x E RN_m}). We let x be the local inverse of F. We claim that x is the desired map for the lemma. We must show that = (0,x) and that V(Xk) = 0 for j m and m + 1 k N. Clearly, x(O,x) = (0,x) because is the inverse of F and F(0, x) = F(0, x) = (0, x). and

1

The Frobenius Theorem

56

Next we show that Lm(Xk) = 0 for m + 1 < k < N. From (5), we have Xs'(0/Otm). Therefore

Lm

TmI

Li

\

—I

a }

=0 if m+1
From the initial condition in (5), we have x = already mentioned, each vector field 12, 1 Therefore, (4) implies

1 k

on the hypersurface M0. As m — 1, is tangent to M0.

L'Xk=O onM0 m+1
=0 m+1
rem.I

4.2

The analytic Frobenius theorem

The complex analytic version of the Frobenius theorem requires the following lemma. LEMMA I

Suppose 1 m n and suppose

has coordinates ((,z) with (

Ctm,

cn-rn

where each

origin in

is a holomorphic function defined in a neighborhood of the Then there is a neighborhood U of the origin in and a

The analytic Frobenius theorem

57

biholomorphic map

with

L3{Zk}=O on U and

Z(O,z)=(O,z)

for

The proof of this lemma is similar to the proof of the corresponding lemma in Section 4.1. The only major change is that instead of solving a system of ordinary differential equations, we must solve a system of partial differential equations. For example, when m = 1, the following system of partial differential equations replaces (3) in Section 4.1: OF1

OF2

=

1

=

= and

F(O,z) = (O,z),

z E C'.

Here, the coordinates for C and z E Since each A3k is holomorphic, a locally defined holomorphic solution F can be found by using the Cauchy—Kowalevsky theorem (see [Jo] or [Fofl. The rest of the proof is the same as the proof of Lemma 1 in Section 4.1 except that the holomorphic version of the inverse function theorem must be used instead of the real version of the inverse function theorem. We leave the details to the reader.

This lemma implies the analytic version of the Frobenius theorem in the same way that the lemma in Section 4.1 implies the real Frobenius theorem. First, we make two definitions. A vector field ak(O/Ozk) in is called a holomorphic vector field if each a, C is holomorphic. A holomorphic or analytic subbundle of dimension m is a subbundle of T"° (Ca) that is locally generated over by m-holomorphic vector fields that are :

linearly independent.

The Frobenius Theorem

58

THEOREM 1

THE ANALYTIC FROBENIUS THEOREM

Suppose L is an rn-dimensional holomorphic, involutive subbundle ofT"° there is a neighborhood U of and a biholomorThen given a point p0 E phic map Z = Z(U) C C'1 such that :U

L{Z3}=Oon Uforrn+l i
is tangent to the rn-dimensional complex

manifold

Thus, the above theorem foliates U into complex manifolds rn so that L, for each p E U.

4.3

of dimension

Almost complex structures

In the analytic version of the Frobenius theorem, the subbundle L is defined on an open subset of complex Euclidean space. This theorem has an abstract version, which we shall state but not prove. First, we make a definition. DEFINITION 1 Let M be a smooth manifold (not necessarily imbedded in (M) (the complexified tangent bundle Let L be a smooth subbundle of of M). We say that the pair (M, L) is an almost complex structure if 1.

LflL={O}

2.

An almost complex structure can only exist if the real dimension of M is T"°(M) is an

even. Note that a complex manifold M together with L =

example of an almost complex structure. From Lemma 3 in Section 3.2, specifying L in the above definition is equivalent to specifying a complex structure map —' T,,(M) which varies — smoothly with p e M. If M is a complex manifold then L = T°" (M) is involutive. This follows by observing that a Lie bracket between any two vector fields that only involve is again a linear combination (over of

Here, (z1,... ,

is a set of local holomorphic coordinates for M.

Similarly, L = T"°(M) is also involutive. A natural question is to ask if the converse is true: if (M, L) is an involutive almost complex structure, then does there exist a complex structure for M so that L = T°"(M)? The answer is yes, locally, and this is the content of the following theorem.

Almost complex structures

THEOREM 1

59

NEWL4NDER-NIRENBERG (NNJ

Suppose (M, L) is an almost complex structure and suppose L is involutive. Then near any given point P0 E M, there is a complex structure for M which makes M a complex manifold so that L = T°" (M).

The proof of this theorem involves a lot of machinery that would take us too far afield and so we only give a reference (see Section 5.7 in [Ho]). The reader who is not familiar with this theorem can take comfort in knowing that it will only be used in Section 10.1 where we discuss Levi flat CR manifolds. If the reader restricts his or her attention to imbedded Levi flat CR submanifolds of then the reader will only need the following imbedded version of the Newlander—Nirenberg theorem whose easy proof we provide. THEOREM 2

Let J be the usual complex structure map on of such that =

Suppose M is a smooth for each p E M. Then M

is a complex submanifold of C'. The proof of Theorem 2 does not follow from the complex analytic version of the Frobenius theorem in the preceding section because we do not know that L = Tc(M) fl is an analytic subbundle of (M is only assumed to be smooth). Theorem 2 does follow from Theorem 1 because L is an involutive, almost complex structure. However, we give a proof that does not require the use of Theorem 1.

We shall show that near any fixed point E M, we can exhibit M as the graph over its tangent space at of a holomorphic mapping. By a translation, we assume the point p,, is the origin. Then we use a complex linear map so that the tangent space of M at 0 is the copy of given by {(0, w) C'; w E } where 2k = dima M. This is accomplished as follows. Since To(M) is J-invariant by hypothesis, the orthogonal complement of To(M), To(M)1 (using the Euclidean metric for is also J-invariant, in view of Lemma 1 in Section 3.2. In addition, J has no real eigenvalues. Therefore, there is a real basis for of the form PROOF

v1,Jv1,... ,Vn_k,JVn_k and

a real basis for To(M) of the form

Vn_k+1,JVn_k+1,...,Vn,JVn. Give 1R2" the coordinates (x1,y1,...

Recall

The Frobenius Theorem

60

Define the linear map A:

by setting

by (real) linearity. Note that A(Jv3) = JA(v3), Viewed as a map from 1 j to C's, A is complex linear. In particular, A is holomorphic and so A{M} also satisfies the hypothesis of Theorem 2. The real tangent space of Let W1 Xn + Xn_k+1 + 2Yn—k+I,— , A{M} at the origin can be identified with A to all of

n. Therefore AoJ = JoA on

Ck}.

{(O,w) E Cn_k x Ck;w

Therefore, near the origin, A{M} is the graph of a smooth map h : Ck ....., Cn_k, i.e.,

A{M} = {(h(w),w);w E Ck}. To show A{M} is a complex submanifold of C'2, we must show h is holomorphic. This is equivalent to showing

h1(p)oJ= for

each p

R2k

Ck. Here,

Joh1(p)

(p) is the push forward map of h at p (see

Chapter 2) where we view h as a map from 1R21C R2n_2k. The J on the left side of the above equation is the complex structure map for R21C, whereas the J

on the right is the complex structure map for Define H : x by

H(w)=(h(w),W) Let L be a vector in To(R21c). For p E R2k

WECk.

for

Ck near 0, we have

=

x

E

=

Since H is the graphing map for A{M} over R2k, is J-invariant, TH(p)(A{M}). Since to Now any vector in

is

belongs to also belongs of the form

W) W

for J{H. (p) (L)}, This establishes h is holomorphic, as desired. I

we see that W = J(L) and therefore h1(p)(JL) = Jh.(p)(L).

h. (p) o J = Jo

5 Distribution Theory

The discussion of solutions to partial differential equations is facilitated by the language of distribution theory. In this chapter we define distributions and discuss the basic operations with distributions. We then discuss a version of

Whimey's extension theorem. The chapter ends with the computations of a fundamental solution for the Cauchy—Riemann operator on C and a fundamental solution for the Laplacian on RN.

5.1

The spaces V' and

('

be an open set in RN. In Section 1.1, we defined to be the space of smooth, complex-valued functions on ft This space is topologized by stating that a sequence n 1,2,.. .} converges to f in provided converges to uniformly on compact subsets of for each multiindex This topology on can also be described by the following seminorms. Let n = 1,2,. . .} be a nested sequence of compact subsets of that exhaust { Let

=ft

:e(cl)

1,2,...aredefined

by

= sup ID'V(x)I

for

xE K,,

fE

IQIn These seminorms satisfy the following properties:

(i)

=

for

E C,f E

(ii) (iii)

each n = 1,2

It is a routine matter to show that

is complete under

61

Distribsuion Theory

62

this topology. A complete topological vector space whose topology is defined

by a countable collection of seminorms is called a Fréchet space. In Section 1.1, we also defined V(1l) to be the space of elements in E(Q) n = 1,2,.. .} is said to converge to with compact support. A sequence in if there exists a fixed compact set K that contains the support of each uniformly on K for each multiindex and such that The space is not a Fr&het space. Using coordinate charts, the above definitions of the topologies of and carry over to the case where is an open subset of a smooth manifold. Most of the time in this chapter, our discussion will be focused on Rh'. However,

much of what is to follow (with the exception of the convolution operator) generalizes to the manifold setting. We leave these routine generalizations to the reader. We now define the space of distributions. DEFINITION 1 For an open subset of (or a smooth man jfold), the space of distributions is defined to be the dual space of D(1l). That is, is the space of all continuous linear maps from D(fl) to C. is defined to be the dual space of 6(11). Note that 6'(Il) is a subspace of D'(Il) because D(11) C 6(11) and since a sequence that converges in D(1l) also converges in 6(11). The pairing between an element T of V'(Il) and an element of V(11) will be denoted by

E C.

Example 1 Let T be a locally integrable function on 11. Then T can be viewed as an element of D'(IZ) by defining (T,

=

f

for

4 E V(1l).

xE(1

If

in D(1l) then clearly (T, (T, Thus, T = (T, defines an element of 2Y(1l). If supp T is a compact subset of 11, then T = (T, is an element of 6'(Il). U

Example 2

For a point p E IRN, the delta function at by the formula

=f(p)

for

defines an

fE6(R").

U

The spaces 7)' and e'

63

Example 3 Suppose M is a smooth, oriented i-dimensional submanifold of RN. Let be Hausdorif i-dimensional measure restricted to M. The measure d/LM is constructed as follows. Let cia be the volume form for M, which is the unique

positive i-form on M of unit length. To construct du, we first focus on a coordinate patch U C M corresponding to an oriented coordinate chart x: U

R'. On U, let

dC&X(dtAAdt) where a is a smooth function on U chosen so that Idol 1 (here denotes the pointwise Euclidean norm as extended to forms (Section 1.5)). The volume form da is then globally constructed on M by covering M with a collection of coordinate patches corresponding to oriented coordinate charts together with a partition of unity. For a compactly supported continuous function g: RN —i C, define I

I

A(g) Note

that A is linear and A(g) 0 for g 0. By the Riesz representation

theorem, there is a unique positive Borel measure d/LM that represents A, i.e.,

A(g) = J9dIzM. As a measure, dpM acts on elements of D(RN) via integration. Hence, d/LM is

an element of DI(RN). If M

is

compact, then

defines an element of

U

Sometimes, we shall emphasize the "variable of integration" on

(T(x),

= (T,

for T E D'(Il),

by writing

E

This notation is motivated by Example 1 above where T is an function and the pairing (T, is given by integration. However, we shafi sometimes use this notation even when T is not a function defined on in the traditional sense.

A distribution T in V'(Il) or £'(cl) is said to vanish on an open subset U of Il if (T, = 0 for all E D(U). The support of a distribution T (denoted supp T) is defined to be the smallest closed set in outside of which T vanishes. In Example 1 above, the support of T as a distribution is the same as the usual support of T as a function defined on Il. In Example 2, supp {6,} is the single point set {p}. In Example 3, supp d/LM is M. In all three examples, if supp

T is compact in Il, then T defines an element of coincidence, as the following lemma shows.

This is not just a

Distribution Theory

64

LEMMA I

If T is an element of e'(cl), then there is a compact set K C M > 0, and a constant C > 0 such that sup

IalM

an

integer

forall fEe(cl).

xE K

In particular, supp T C K. Conversely, if T is an element of D' (Il) and supp T is compact in Il, then T defines an element of e'(cl).

It is a standard fact from topology that a continuous linear complexvalued map defined on a Fréchet space must be bounded in absolute value by a constant multiple of a finite sum of seminorms. Thus, the above inequality PROOF

follows for some integer M, constant C, and compact set K. In particular, supp T must be contained in K, for if not, then there must exist an element

f

K) with (T,

0. This contradicts the above inequality. For the converse, suppose T belongs to D'(cl) and supp T is compact. Let

e V(1l) be a smooth cutoff function (see Section 1.1) with neighborhood of supp T. Define

= 1 on a

f

=

The right side is well defined because çbf belongs to in 6(e), then in to D'(fl). Also, if this definition of T makes it a well-defined distribution in vanishes on a neighborhood of supp T, clearly

f

foreach

and T belongs Therefore, Since 1 —

fEV(fl).

Hence, the above definition of T as a linear map acting on 6(11) agrees with the original definition of T as a linear map acting on D(11) C 6(11). I

spaces V'(11) and 6'(ll) are endowed with a topology called the weak n = 1,2, . topology. This is defined by declaring that a sequence converges to T in V'(11) (or 6'(11)) if The

.

for each in D(1l) (or 6(11)). This convergence is not required to be uniform As an example, if in is a sequence of continuous functions on 11 that converges to the function T uniformly on compact subsets of 11, then the sequence

also converges to T as distributions in D'(11). However, the topology on

v'(cl) allows for much weaker types of convergence. For example, define the sequence e€(x) = e(x/f)f_", for > 0, where e(y) = This sequence was used in Section 1.1 for the proof of the Weierstrass theorem. Using the ideas from its proof, the reader can show e( m 2)' (Rh') as —p 0. Note that —p o for x 0 (but not uniformly) and eE(0) oo.

Opemlions with dislrlbWlons

5.2

65

Operations with distributions

All of the definitions of operations with distributions are motivated by considering the case where the distribution is a smooth function.

Multiplication of a distribution by a smooth function If T and

are

smooth functions on

then T• i11) defines a V' (11)-distribution

by

=

f

E D(1l).

for

So if T is an arbitrary V' (11)-distribution and Note that (Tm, = (T, by if belongs to 6(11), then we define the distribution

=

for

E D(1l).

With this definition, satisfies all the required properties for a distribution. Moreover, this definition agrees with the action (as a distribution) of Til' when both T and are smooth functions.

Differentiation If T is a smooth function on 11 and

=

f

E D(11), then

=

f

T is an arbitrary D'(Il)-distribution, then we define the distribution

by

= If

q5

in D(fl) then the sequence

is continuous and so Thus, As an example, we have

=

in D(1l). also converges to is a well-defined V' (11)-distribution. for q5 E 6(RN).

If a sequence of distributions } in V' (Il) converges to the distribution This follows from the then DaTa also converges to T as n —p oo, definition of the weak topology on D'(Il). This is in contrast to other types of topologies on function spaces where this does not hold. For example, the is not even sin nx converges uniformly to 0 but = sequence pointwise convergent.

Theor,

66

Convolution

=

Here, we specialize to the case

IRN.

If T belongs to e(RN) and t& belongs is the smooth function defined by

to D(RN), then the convolution T * (T *

Ø)RN

=

f

VERN

f(f - Y)ø(X)dx) dy.

T(y)

With the notation

T(y)i,i'(z — y)dy.

we have

As a distribution on

(T *

=

=

this becomes

=

J T(y)(il'*Ø)(y) VERN

= and belong to D(RN). Therefore, well defined for T in Ds(laN). If T is an arbitrary DI(RN distribution and if ii' E D(RN), then we define Now

(T,

belongs to D(RN) because both

* 65)RN

is

=

for

D(RN).

inV(RN). Therefore,

Øin

is

a well-defined element of Dl(RN). From the above discussion, this definition agrees with the action of T * as a distribution when T is a smooth function. * then belongs to and ó belongs to If belongs to

So the same definition can be used to define the convolution of an ('(R' )-distribution T with an element v of ((Rh'). The result is a V1(RN distribution.

Let us examine the definition of T * t more carefully. If T is an element of and 0 are elements of V(Rv), then

P!(Rv) and

(T *

= (T.

*

= (T(y).

f v(r -

z€Rs

/

Operations with distributions

67

As already mentioned, the above notation indicates that the distribution T pairs with the function — in the y-variable. The term — y)Ø(x)dx can be approximated by Riemann sums in x. Moreover, this approximation can be accomplished in the )-topology (in the y-variable). Therefore, using the linearity and continuity of T, we can interchange the order of the pairing of T with the integral in x to obtain (T *

- Y))YERN

= J (T(y),

Ø)RN

XERN

= ((T(y), Therefore, the distribution T *

,

is given by pairing with the following function:

(T(y), i,b(x

x

—

—

This formula generalizes the usual convolution formula when T is a smooth

function on RN If is an element of D(RN), then a difference quotient for converges in V(IRN) to the corresponding derivative of So if T E DI(IRN), then

(T(y),

* i1b}(x)

—

This shows that T *

is a smooth function on RN. By the definition of the derivative of a distribution, we also have = (DAT,

*

—

can be written as either T * is summarized in the following lemma. So

Y))YERN.

or

*

*

This discussion

LEMMA I

Suppose T is an element of D!(RN) and is an element of distribution T * is given by pairing with the smooth function T*

= (T(y), D°T *

=T* continuous linear map from V(RN) to Moreover,

*

—

Then the

y))YERN.

The operator

i—s

T

*

is a

We can also define the convolution of a 2)'-distribution with an e'-distribution by mimicking the above definition of the convolution of an element of V' (RN)

Distribution Theory

68

with an element of V(RN). First we define 't for T E V'(R1") by

for

= T(—x) when T is a smooth

This definition agrees with the formula function. For T E and S E

the distribution T * S is defined

by

for

is a smooth function (Lemma 1) with compact support, the right side is well defined. The same definition can be used if T is an )-distribution and S is a Since

*

LEMMA 2

For T E 1Y (Rh'),

*

T = T.

PROOF We have for

= —

(by Lemma 1)

Y))YERN

= (T(y),

(by Definition of

= Hence, öo *

T = T and the lemma follows.

I

If T is a smooth function on RN, then Lemma 2 can be established more directly by using Lemma 1 ((5o

* T)(x) = (öo(y), T(x —

T(x).

Lemma 2 shows that convolution with öo represents the identity operator. This will be important in Section 5.4 when we discuss fundamental solutions for partial differential operators with constant coefficients. THEOREM I

For an open set

C Ri', D(1Z) is a dense subset of V'(cZ) with the weak

topology.

Let T be an element of of compact sets in Il. Let

by an increasing sequence be a cutoff function that is 1 on

Exhaust

PROOF

E

Operations with distributions

69

converges to T in D'(Il) as n oc. So Clearly it suffices to approximate any element T of 6'(Il) by a sequence of smooth functions on Il' where is a neighborhood of supp T. Let > O} C be the mollifler sequence defined in Section 1.1. It follows from the ideas in * 4) Section 1.1 that as 0, provided 4) is an element of 4) in a neighborhood of

in

is a smooth function whose support is contained From Lemma 1, T * provided 0 is suitably small. From the definitions, we have for

4)

Therefore, T *

T in e'(cl'),

as

desired.

e(cl').

4) in

I

Tensor products

Suppose T is an element of Dl(IRN) and S is an element of V/(Rc). We wish

to define the tensor product T ® S E

in such a way that the

x

following holds:

((T® for 4)

V(IRN),

=

(T,Ø)RN

.

E V(IR"). Let g be an element of V(RN x

for XERN

(1)

The function (2)

is smooth with compact support. This follows from the fact that S is linear and that difference quotients of g(x, y) in the s-variable converge in D(RN x to the corresponding s-derivative of g(x, y). Therefore the pairing

(T(x), (S(y), g(x,

)XERN

(3)

x then the corresponding sequence of functions defined in (2) with g replaced by gn is a convergent sequence in D(R"). So, the corresponding sequence of complex numbers in (3) also converges. We define (T ® 5, g)RN by (3). By the above discussion, T ® S is a well-defined element in D#(RN x Rk). If g(x, y) = 4)(x)ib(y), then (3) reduces to (1), as desired. Instead of (3), we could have used

is well defined for T E 2Y(RN). Ifg,2 —+ g in

(S(y), (T(x),g(x, Y))XEaN )yERk

(4)

as the definition ofT® S. This formula also yields (1) in the case g(x,y) = So the expressions in (4) and (3) agree for functions of the form Since the set of functions that are finite sums of terms g(.r. y) =

Distribution Theory

70

D(RN x Rk) for E of the form by the Weierstrass theorem (see Section 1.1), clearly (4) and (3) agree for all g E D(RN x As an example, consider the tensor product of in DI(RN) with the function I as an element of 1Y(R") For g E D(RN x Rk), we have ® 1, g(x, Y))(x.y)ERN

= (6o(x),

(1(y), g(x, Y))YERk )XERN

y)dy

/ =

f

g(O,y)dy.

YERk

Composing a distribution with a diffeomorphism —* is a diffeoSuppose and are connected open sets in RN and F: morphism. If T is a smooth, complex-valued function on Il', then T o F is a smooth function on ft As a distribution on Il, we have

(To F,

=

f

=

f T(x)Ø(F' (x)) I det DF' (x) dx

for

0E

by the change of variables x = F(y). So

(T0F,0)n = (T,(OoF')IDet DF'I)c'. F is a diffeomorphism, Det DF' is either always positive or always DF'I is a smooth function with compact So (0° support in el'. We therefore define, for T E D(1l'), the distribution T o F E Since

negative on

D'(fl) by the above formula. As an application of these last two operations, we show how to extend a distribution defined on a submanifold of RN to a distribution defined on all of Suppose N = e+ k where Lis the real dimension of the submanifold in RN. By a partition of unity, we can localize the problem. By a coordinate change and the above change of variables formula, we may assume that the submanifold is the copy of Rt given by {(x,O) E Rt x Rk;x E A distribution on can then be extended to all of R1 x by tensoring it with the function I on Rk.

Whitney's extension theorem

5.3

71

Whitney's extension theorem

In its simplest form (due to Bore!), Whitney's extension theorem states that given

any infinite sequence of real numbers {ao, a1,...), there is a smooth function for n = 0,1 Here, f(n) is the nth derivative f: R —' R with = of f. We wish to prove this theorem along with its natural generalization to higher dimensions. First, we need a preliminary result from distribution theory which is interesting in its own right. THEOREM I

Let fl be an and y Let (x, y) be coordinates for RN with x = {(x,0) 1l;x Rk}. Suppose T is an element open set in R" and let with support in Then there is an integer M 0 and a collection of with M} such that of distributions {T0 e'(clo); a = (a', .. . ,

T(x,y)= aI<M

is all of RN and leave to the reader the minor We assume that modifications required in the case where fl is not all of RN. Since supp T C {y = 0}, by Lemma 1 in Section 5.1 there is a compact set K in RN, a constant C > 0, and an integer M 0 such that PROOF

l(T,f)RNI C sup

for

I E e(RN).

(I)

IczIM (x,O)EK

is a partial differential operator in both x and y of order al S M. For a = (ai,..., aNk) with al <M, define the distribution T0 eI(Rk) by Here,

(Ta(s),

e(Rk).

for

=

We claim {TQ } satisfies the requirements of the theorem. its Taylor expansion in the y-variable is Given f

f(x,y)= V31M

The Taylor remainder e satisfies 0)

=

0

1131

S

M.

So in view of (1) (T,f)RN =

.ij. I/31M

.

(x,y)ERN

(2)

Diiiribution Theoiy

72

On the other hand, from the Taylor expansion of f, we have

1

1

\IaIM

RN

= xER"

IoIM Clearly —

f0 =

if Therefore ®

= By the definition of

the right side equals

(z,y)ERN

By (2), this equals (T, f)RN. This completes the proof of the theorem.

I

Now we prove Whitney's theorem. THEOREM 2

Suppose RN has the coordinates (x, y) with x E and y E RN* Let be an open set in RIC. Suppose is an element of e(110) for each index = (cr1,... , with 0, j N — k. Then there is a function x RN_Ic) such that for each IE 1

(x,0)=a0(x) for The function

f is unique modulo the space of smooth functions on x {0}.

X

which vanish to infinite order on

PROOF The uniqueness assertion is obvious. So we shall concentrate on the existence part of the theorem. As in the proof of Theorem 1, we shall assume is all of Rk. if N = I and k = 0, then the theorem reduces to showing that for a given sequence of real numbers {ao, ai,. . .}, there is a smooth function f: R —' R

Whitney's extension theorem

73

with

forn=O,1. This

can be done explicitly by first choosing a cutoff function 0

E

D(R) with

iflyll

1

10

iflyl

2.

Then we let (3)

= There

exist

(depending on

I) so

that for any integer a 0

C is a uniform constant depending only on a. Therefore, the sum in (3) converges in the topology of e(R). Since 0 in a neighborhood of the for n 0, 1,2,..., as desired. origin, = The reader can modify this proof so that it will work in the general case. where

1

However, a slicker proof is available with the help of a little functional analysis. We consider the space

fle(Rk) where fl0 denotes the infinite cartesian product indexed by a (as, . .. , where each a1 is a nonnegative integer. Elements of this space are infinite tuples

(a0) where each a0 is an element of e(lRk). This space is given the product topology, that is, a sequence (az) for n = 1,2,... is said to converge to (a0) in as n —' oc if for each fixed a0 in e(Rk). This makes 110 e(lRk) into a Fréchet space (since each e(Rk) is a Fréchet space and since the index set {a = (aj,. . ,aN_k);a1 is a nonnegative integer} is countable). x RN_k) Define the map by fi .

(irf)0(x) =

1 01°1f

(x,0)

for each

a.

We want to show the map ir is surjective. By looking at the subspace of polynomials on RN, it is clear from the definition of the topology of fl0 e(Rk) that the image of ir is dense. So it suffices to show the image of ir is closed. By the closed range theorem for Fréchet spaces, it suffices to show the range of the dual to ir ir':

{He(Rk)}

74

Dtstribwion Theory

is closed. Here, ir' is defined by

(lr'(T),f)RN = (T,irf)

forTE {fl e(Rk)}'

and

f E e(IRN). We have

el(lRk) are infinite tuples (Ta) With where, by definition, the elements of eF(Rk) such that only a finite number of the T0 are nonzero. An element Ta E (Ta) in E061(Rlc) acts on an element E by

((Ta),

=

fa)Rk.

The sum on the right is well defined since only a finite number of the Ta

are

nonzero.

We now compute ir'. We have

(lr'{(Ta)},f)RN = ((Ta),irf)

=

laif

/

(Ta(x)

®

Therefore

(_l)1k1

&

( ) ôyOaY

which has its support in the set 11o = {(x,o) E E R"}. This equation together with Theorem 1 implies that the range of ir' is the set of all elements RN_k) whose support is contained in in eF(RIc This set of distributions is closed with respect to the weak topology on el(Rk x RN_k). As mentioned earlier, the closed range theorem implies that the range of ir is closed. Since ir also has dense range, ir must be surjective and so the proof of the theorem is complete.

I

54 Fundamental solutions for partial differential equations As mentioned at the beginning of this chapter, one of the reasons for the introduction of distribution theory is that it provides a convenient language to discuss

Fundamental solutions for partial differential equations

75

solutions to partial differential equations. We start by defining a fundamental solution for a constant coefficient partial differential operator

P(D)= I

a0EC.

T E IY(RN) is a fundamental solution for P(D) if

P(D){T} = The reason for the name "fundamental solution," is that a solution to the equation P(D){u} = for E D(RN) can be found by convolution with T,

as the following theorem shows. THEOREM I

Suppose P(D) is a partial differential operator with constant coefficients. Supis a E D(IRN) then u =

pose T is afundamental solution for P(D). If solution to the differential equation P(D){u} = PROOF

From Lemmas 1 and 2 in Section 5.2, we have

= (P(D){T})

P(D){T * Therefore,

P(D){T *

= 0, as desired.

*

=

*

=

I

We remark that if P(D) has variable coefficients, then this theorem is not true. This is because the step P(D){T * = (P(D){T}) * is not valid; for in order to apply the derivatives to T, an integration by parts is required and expressions involving the derivatives of the coefficients of P(D) will appear. In Part IV, we shall need fundamental solutions for the Cauchy—Riemann operator on C and the Laplacian on RN.

THEOREM 2

The distribution T(z) = /irz is a fundamental solution for the Cauchy-. Riemann operator = l/2(ô/ôx + iO/Oy) on C. 1

Note that T(z) = l/irz is a locally integrable function on C and so T defines an element of D'(C). To prove this theorem, we must show PROOF

all) —

az

iirzj

=

Distrthutkn Theory

76

which is equivalent to —

for

q5(O)

(1)

zEC

This is a generalized version of the Cauchy integral formula which is established

R < oo, let

by the following safety disc argument. For 0 <

C;€ < Izi
= {z

Here, R is chosen so that supp C {IzI
[

—

0.

1 1 2iriz

which means The curve is oriented as the boundary of the open set is oriented in the counterclockwise direction and {IzI = R} that {IzI = €} is oriented in the clockwise direction. Since d = 0 + 8 and since =0 on {IzI = R}, we obtain —

£ 7

—

2iriz

—

{IzI=€}

if[

2iriz

A dz = 2idx A dy, this becomes

Since

—

,(

—

2iriz

1'

{IzI=e}

As

0,

iff

the right side converges to

if

dxAdy

C

the integrand is locally integrable on C. By parameterizing {IzI = and using the continuity of at 0, the left side converges to I 0. This proves (1) and completes the proof of the theorem.

because

by

z=

as

eett

THEOREM 3

Let

I I.

where

—

ifN=2

'N>3 —

i = (2irN/2/r(N/2)) is the volume of the unit sphere in RN. Then

T is a fundamental solution for the Laplacian

=

on RN.

Fundamental solutions for partial differential equations

77

For N = 2, we have

PROOF

0(1 i9z

1

zEC

4irz

2ir

in the sense of distribution theory. This follows from straightforward calculus if 0. Since 1 /irz is locally integrable, the above equation holds as distributions z in a neighborhood of the origin as well. Theorem 3 now follows from the above 4(02/OWz). equation together with Theorem 2 and the fact that

For N 3, we use a safety disc argument similar to the one in the proof of Theorem 2. We must show

f

=

for

E

RN;e < lxi


D(RN).

(2)

XERN

For

R< oc, define

0<

= {x

< R}. This time we apply Green's After formula (see Corollary 2 in Section 2.5) on with u = T and v = (by explicit computation), we obtain noting that = 0 on

where R is chosen with supp

J

=

C

J

—

Now = 0 on a N is the unit outward normal vector field to neighborhood of {lxl = R} by the choice of R. Therefore, the integral on the right is just an integral over {lxl = €}. The above equation reduces to where

J —.

0.

T is locally integrable on RN, the left side converges toJRN For the right side, note that IT(x)I C is a uniform constant. Thus

J T(x)NØ(x)da(x)

as

N-I

(4)

Distribution Theory

78

as e —+ 0.

In polar coordinates, N = —(0/Or) on {IxI = f}. So

NT(x)

—(2— N)_Iw

=

Therefore, we have

-

f

f

{lzI=}

as e right side converges to with (3) and (4) show that (2) holds and the theorem follows.

By the continuity of

the

0.

This together I

6 Currents

Roughly speaking, currents are to forms what distributions are to functions. Therefore, much of this chapter parallels the previous chapter on distributions. However, we leave to Part IV the discussion of fundamental solutions for various

overdetermined systems of partial differential equations acting on differential forms.

6.1

Definitions

We consider two spaces of forms. For an open subset

of a smooth manifold,

let

be the space of smooth differential forms of degree r

yr

be the space of elements in er

with compact support.

is topologized as follows. In local coordinates, x = (Si,. .. The space RN, a sequence n = 1,2,..., can be written UC E

,

= =r where each

is an element of 6(U). We say that

converges to f =

II=r f'dx' in the coordinate patch U if the component functions

converge

to f1 as n oo in the topology of 6(U) (i.e., uniform convergence on compact is said to converge to the sets of each derivative). The sequence E converges to / in each coordinate patch U of ft This form / in if definition of convergence is independent of the choice of coordinate charts. n = is defined analogously. Here, a sequence The topology of yr 1,2,... is said to converge to in DT(1l) if there is a compact set K C with supp C K for n = 1,2,... and each derivative of the coefficient functions converge to the corresponding derivative of the coefficient functions of of

79

Currents

80

We now define the space of currents as the dual of the space of forms.

For an open subset of a smooth manifold, the dual space is denoted by {7Y (Q) }' and it is called the space of currents of of D' is denoted by {e"(cZ)}'. dimension r. Likewise, the dual of the space

DEFINITION I

is the space of all continuous complex linear funcThe dual space of and since the Since is a subspace of tionals defined on inclusion map is continuous, clearly {6'(Iz)}' is a subspace of {7Y'(IZ)}'. is denoted The pairing between elements of {D"(fl)}' and elements of for distributions). Occasionally, we shall (as opposed to ( , by ( , emphasize the variable in by writing

(T(x), f(x))XEn = (T, f)cz

for

TE

fE

The following examples parallel the examples of distributions given in the previous chapter.

Example 1 be a form of degree N — r and let T = Let Q be an open subset of (i.e., = N — r). Suppose Tj: —p C is locally integrable. T is viewed as a current in

by defining

f T(x)Af(x)

f is an N-form on

for

with integrable coefficients and so the right

C

side is well defined.

If f = fjdx',

= r, for some function fj

(0 where

f

then

Tj(x)fj(x)dx

is defined by

=

A...AdXN.

Tjfjdx is the same as the pairing between the distribution Tj E In this way, currents are closely related to with the function f' E [1 distributions. This relationship will be made precise in Lemma 1. Note that

Example 2 Consider the point p E RN. We view [p] as a current in {eo(RN )}' by defining

([p],f)RN = f(p)

for

f E 60(RN) = e(RN).

Definitions

81

This example is analogous to the delta function at p given in the last chapter. I]

Example 3

Suppose M is an oriented submanifold of RN of dimension r. We view [M] as a current in {1Y(M)}' by defining ([M),f)RN

for

IEVT(IRN).

If M is compact, then we can allow I

to be in er(RN) and so in this case, [M] is an element of {er(RN )}'. Note that Example 2 is a special case of

Example 3.

0

The last two examples illustrate the reason the space is called the space of currents of dimension r. A basic example of such a current is given by integration over an r-dimensional submanifold.

Since these three classes of examples parallel the three classes of examples of distributions from the previous chapter, there should be a connection between distributions and currents. Let be an open subset of an oriented mamfold of dimension N and let 0 q N. We let be the space of q-forms whose coefficients are D'(IZ)-distributions. This space will be called the space of currents of degree q. Likewise, we let be the space of q-forms whose coefficients DEFINITION 2

are 6' (11)-distributions. LEMMA

1

Suppose

11 is an open subset of an oriented N-dimensional man(fold and let N. Then {Dr(ll)}I is isomorphic to

0

r

PROOF We concentrate first on the case where 11 is an open subset of RN. The idea of the proof comes from examining Example 1 above. For T = E

jyN_r(Q)

and

f = fjdx1 E 7Y(c), we define T as a current in {1Y(11)}' by

setting 1

0

if

where as in Example 1, is defined by dx' A = Since Tj is a V'(ll)-distribution and fi is an element of V(cl), clearly the right side is well defined. Therefore, any current of degree N — r can be considered as

an element of {rr(cl)}'.

Currents

82

Conversely, suppose T is an element of {D' }'. For any increasing multiindex J of length N — r, define the distribution Tj E D'(Il) by for

where J' is the increasing multiindex of length r which is comprised of the indices in { 1, . , N} that do not belong to J. It is then an easy exercise to . .

show

T=

u

The proof of Lemma 1 for oriented manifolds is the same. In this case, the form dx is replaced by the volume form do' (constructed as in Section 5.1 with some suitably chosen metric replacing the Euclidean metric on RN). Let us describe the above three examples from the degree point of view. The first example is already presented as a form with distribution coefficients. For Example 2, we have [p] E eIN(RN). For Example 3, we specialize to the case where M is the smooth boundary of an open set C RN with the usual boundary orientation. Suppose ci {x E Rh'; p(x)
[M] = /IMdP

which exhibits M as a degree 1 current. To see this, we first need a working formula for /1M. If E V(RN), then from Section 5.1,

where

do' is the volume form on M. Since IVpI = I on M do' =

Let g = ódx1 A.. .Adx3 A..

with

From the definitions,

0E

we have (/2Mdp,g)RN

=(_l)J

p VXj

RN

(I) On the other hand

(Vpjdp)g

g

=

(since IVpI = 1)

A g) + dp A (Vpjg)

DefinUions

83

where the last equality follows from the product rule for (Lemma 1 in Section 1.5). Since j*dp = 0 on M where j: M —* RN is inclusion, we have ([M],9)RN

([M],g)pN =

which by (1) above proves our claim that [M] =

... = Pd(s) =

In a similar manner, if M = {x E R";p1(x)

O}, then the

current EM] is given by

/LMadPIA...AdPd where a = IdPiA. . .AdpdI' or —IdpiA. .AdpdI' depending on the orientation given to M. As a further example, let = {(x,x) E RN x RN;x E RN}. is called the diagonal of RN x RN.) The current (integration over has dimension N and therefore degree N in RN x RN. From the degree point of view, = A... A d(xN — YN) where 6o(x — y) is the distribution on öo(x — y)d(xj — x RN defined by .

(6o(x

—

y), Ø(x, Y))(x,y)ERN XRN

=

f

x)dx.

XERN

We leave the verification of this to the reader. The definition of the weak topology for {17(1l)}' is analogous to the definition of the weak topology for D'(Il). A sequence T in {Dr(1l)}I if —' for each form g E This is equivalent to the following characterization of convergence using the degree point of view: a sequence of currents

III=N—r,

for

n=1,2,...

converges to T1dx1 if and only if T' in D'(Ifl. Since V(1l) is a dense subset of is a dense subset of We should say a few words about currents on a complex manifold. For an open subset of a complex n-dimensional manifold, recall that (Il) is the space of smooth compactly supported forms of bidegree (p, q) (0 p, q S n). Its dual space is the space of currents of bidimension (p,q). By definition, the space of currents of bidegree (p, q) (denoted (Il)) is the

Currents

84

space

of (p, q)-forms with coefficients belonging to 1Y(Ifl. The analogue of

for this case is that {DP'Q(cZ)}' is isomorphic to All of these statements also hold with V replaced by e. Recall there is a splitting of an r-form on a complex manifold into its various bidegrees. We have Lemma

1

v"(cI) =

e DO.r(I1)

e

The same applies to currents, i.e.,

...

DFr(Il) =

DIr(cI)

with p + q = r be the natural projection. For a current T, we often write for irPQ{T}. As an example, let = {z E C?z; p(z)
= and

=

6.2

Operations with currents

Analogous to distribution theory, the definitions of operations with currents are motivated by considering the case when the current is a smooth form.

Wedge product of a current with a smooth form If T is a smooth form of degree q on and f is a smooth form of degree r on then T A f is a smooth form of degree q + r. As a current on

(TA f,

=

J(T A f) A

for

=(T,fAq5)c1. We

define T A f for a current T E

and a smooth form f E

the formula

(TA f,

= (T, f A

The result, T A f, is a current in

for

by

Operations with currents

85

Exterior derivative Suppose T is a smooth form of degree N — is an element of eN_r+1(cl) and

r and let 4 E

Then

dT

By the product rule for the exterior derivative (Lemma 1 in Section 1.4), we have

=

J

has compact support in by Stokes' theorem. So Since T A

the first integral on the rigbt vanishes

=

= T for the exterior derivative raises the degree of a current by one and

therefore lowers the dimension by one. Suppose is an open subset of RN. If T = Tjdx1, T1 E 1Y(1Z) then an equivalent expression for dT is given by

dT=

Ox,

= N — r, and

AcLX1

where t97'j /Ox, is the derivative of T1 in the sense of distributions. This formula generalizes the usual exterior derivative formula for smooth forms. As with distributions, if T in {iY(1Z)}' then the sequence converges to dT in As stated in Section 6.1, an important class of currents is the class of submanifolds. So it is natural to compute the exterior derivative of such a current. THEOREM 1

Suppose M is an oriented r-dimensional submanjfold with boundary contained in a smooth N-dimensional manifold X. Then dIM) = [OMJ.

Currents

86

The proof will follow from Stokes' theorem. Suppose Then

PROOF

of

is an element

= = (_1)N_f+1

f

(by Stokes' theorem)

= as desired.

I

of a complex manifold, recall that d = 0 + 0 where

For an open subset

are defined by 0 =

and o d.

=

These same formulas extend the definition of 0 and 0 to currents. From the definition of the exterior derivative of a current, if TE then o d and

=

for

=

E

for

E

= {z E

As an example, suppose is an open set given by IR is smooth with IVpI = 1 on where p: O and dimension 2n. From Theorem 1


d[11] =

Thus

= —[ocI]"°

and

Note that

for

=

E

and

= In words, the pairing between

as the usual integral over

and a form of degree 2n— 1 is the same of the piece of of bidegree n, n — 1.

87

Operations with currents

The push forward of a current under a smooth map —* Il' be and Il' are open subsets of smooth manifolds, and let F: a smooth map. The pull back operator F*: can be dualized to obtain a map defined on currents.

Suppose

Il' be a smooth map. For T Let F: push forward ofT via F, denoted F.T, is the current in DEFINITION I

= (T,F*Ø)c2, for

the defined by

q5

Since F*Ø is an element of the right side of this definition is well defined. Moreover if in then in and so F.T is a continuous linear functional on e"(cz'). Hence, F.T is a well defined (Il) }'-current. The push forward preserves dimension (but not degree) because F preserves degree. Note that F4 does not necessarily have compact support even when 0 has compact support (unless F is proper). For this reason, the push forward of an element in is not well defined (unless F is proper). For smooth maps F and G we have (FoG) = G*oF* on forms. Therefore, we have (FoG). = F. We now give some examples. Suppose M and M' are smooth, oriented RN' be an orientationsubmanifolds of RN and RN , respectively. Let F: RN preserving bijective smooth map with M' = F{M}. Then FP [MI = [M') because

(F.[M],cb)RN' =

for

MJF*0

=

J

(since F is orientation preserving)

=

is the projection ir(x, y) = y x IRk) c on the subspace

As another example, suppose ir: RN x IRk

for x E

Let us compute

+

IRk

Currents

88

{6t(RN

ForTE

x

(ir.T,Ø)Rk

x IRk)and

E Dr(Rk)

= =

=

J

RN xRk

f

J (

VERk

Now the inner x-integral is zero unless all the dx's are present. So we write

T(x,y) =

dx1 AT1(x,y) r=O

I1I=r

each T1 is a form in the y-variables with coefficients that depend on x and y. With this notation, we have where

J T(x,y)=XERNf

XERN

All the x's and dx's are integrated leaving a differential form in y. From the above, we have

=

(I

ERN

yEW'

and so (ir.T)(y)=

for

f T(x,y)

In words, the push forward under ir is the same as the operation of integrating out the fiber of ir. The above idea is illustrated in the next example, which we state as a lemma (to be used in Part IV). LEMMA I

Define r:

{(r(RN

x RN x

RN by

r(y,x) =

x

—

y.

If T E D2N_?'(RN x RN) c

then

f

=

(sT)(y,x)

yE

wheres: RN xRN

XRN is defined by s(y,x)—(y,x+y).

Operations with currents

89

Sometimes, we shall use the notation T(y, x + y) for (8*T)(y, x). The form T(y, x + y) is the differential form obtained by replacing x by x + y in each coefficient of T and by replacing each by dx3 + dy,. On the right side of the equation in the lemma, the only nontrivial contribution in the integral comes from the terms with all the dy's. The y's and dy's are then integrated leaving a differential form in x. PROOF

Suppose

is an element of

Then

= = Now, Det(Ds)

J

RN xRN

= 1 and so s is an orientation-preserving linear isomorphism.

By the change of variables formula for integration (T*T,4)RN

= =

J

RN xRN

f f

rERN YERN

where the last equation follows from

= (ros)s and because (ros)(y, x) =

x. Therefore (T*T,Ø)RN

=

J

/

'yER"

and the proof of the lemma is complete. For

I

each x E RN, note that {s(y,x); y e RN} = r-'(x). So again, we see

that the operation of push forward is the same as the operation of integrating out the fiber. This idea is illustrated in a more general context in the proof of Lemma 5, below. As the next lemma shows, the push forward operation via an orientationpreserving diffeomorphism is the same as the operation of pull back via its inverse. LEMMA 2 Suppose

F:

—+

{(r(Il)}l

are open subsets of oriented N-dimensional man jfolds. Let be an orientation-preserving djffeomorphism. For T E c

and

F,T=

Currents

90

From the proof given below, it will be clear that if F is orientation reversing,

then

=

PROOF

Let 0

be

an element of

Then

=

=

JTAF*0.

Il' —' Il is an orientation-preserving diffeomorphism, the change Since of variables formula for integrals yields A0

= and the proof of the lemma is complete.

I

Since the pull back operator on differential forms commutes with the exterior derivative, the push forward operator commutes with the exterior derivative up to a sign factor.

LEMMA 3

and are open subsets of oriented dimensions N and N', respectively. Let F:

X and Y with real

Suppose

o

as operators from PROOF

dx = (_l)N+N'dy

Il' be a smooth map. Then 0

to

Suppose that T is an element of {6"(Il)}'

and let 0

E

Then

= (dxT,F0)cz =

= = = To see that the minus sign is correct in the last equality, recall that F, preserves

dimension and so F,T has dimension r, or equivalently, degree N' — r as = (....1)N'—r+1 a current on the N'-dimensional set Thus (F,T, dy0)cz' from the definition of the exterior derivative of a current. This completes the proof of the lemma.

I

Operations with currents

91

Now suppose ci and ci' are open subsets of complex manifolds and let ci' be a holomorphic map. From Lemma 3 in Section 3.3, F pre-

F: ci

preserves the bidimension serves bidegree. The dual of this statement is that commutes with the of currents. This together with Lemma 3 imply that Cauchy—Riemann operator. LEMMA 4 Suppose ci and ci' are open subsets of complex manifolds of complex dimensions

n and n' respectively. Let F: ci

ci' be a holomorphic map. Then o

= 00 F,,.

as operators from {6P'Q (ci) }' to Since

(ci') }'.

the real dimension of a complex manifold is even, there is no sign

factor in this lemma.

The pull back of a current via a smooth map Defining the pull back of a current is perhaps a little nonstandard. However,

since the space of differential forms make up a large subclass of currents and since the pull back of a differential form is well defined, we find it convenient to extend (where possible) the pull back operator to currents. Simply put, the pull back operator is defined to be the dual of the push forward operator. To make this precise, we must show that the push forward operator sends smooth forms to smooth forms in a continuous manner. This is clear for the maps and 'r as shown earlier. We shall now prove that this holds more generally. LEMMA 5

Suppose ci and ci' are open subsets of oriented manifolds of dimensions N and ci' be a smooth surjective map N', respectively with N N'. Let F: ci has rank N' (i.e., maximal rank) at each such that point p E ci. Then is a continuous map from 1Y(ci) to

PROOF By a partition of unity argument and by the use of local coordinates, we may assume that ci and ci' are open subsets of RN and RN', respectively. Let po be an arbitrary point in ci. Since DF(po) has maximal rank = N', we may arrange coordinates (x,y) for RN so that x RN_N' and y RN' and so that is a nonsingular N' x N' matrix. Let p0 = (x0, yo). From the inverse x RN' of the function theorem, there is a neighborhood of (xo, F(po)) in form U' xV' with V' C ci' and a diffeomorphism G: U' xV' G{U' x V'} C ci such that F(G(x,y)) = yfory V'. That is, FoG: U' xV' V' is the

projection ir: U' x V'

V',ir(x,y) =

y.

Currents

92

Let 0 be an element of V'(G{U' x V'}). Then

o(G'O). =

G*q5

or (by Lemma 2), is a smooth form with compact support in U' x V'. Since G is a diffeomorphism, the map 0 p—' Since

is a continuous linear map from V"(G{U' x V'}) to V"(U' x V'). In addition, since F o G = ir, the computation of preceding Lemma 1 yields

J

(1)

XERN - N'

where the integral on the right involves the variable x and all the dx's, leaving a

differential form in y. From this expression, it follows that

is a smooth form

with compact support in V' C ci' and that the map p—' F.4 is a continuous linear map from IY(G{U x V'}) to .DN'_N+r(VI). The dimension of F.qS (as a current) is N — r since preserves dimension. Therefore, the degree of

isN'—N+r. The general case for

E 7Y (ci) now follows by a partition of unity argument

subordinate to an open cover of ci by open sets of the form G{U' x V'} as above. The proof of the lemma is complete.

I

GS0 or _G*0, (1) shows that the operation of push forward is the same as the operation of integrating out the fiber (for each fixed y E V', the fiber is the set {G(x,y); XE RN_N })• We now define the pull back of a current. Since

DEFINITION 2

Suppose ci and ci' are open subsets of the oriented manifolds

X and Y with dimensions N and N' (N N') respectively. Suppose F: ci ci' is a smooth surjective map so that

—'

has maximal

rank = N' at each point p E ci. For a current T E V"(ci'), define the pull back FST E D"(ci) by

for 0 E Lemma 5 ensures that FT is a well-defined element of Note that since the push forward operator preserves dimension, the pull back operator preserves the degree of a current. In the case where the current T is given by a smooth form, then F*T as defined above is the same as the usual pull back defined in Section 1.3 or 2.4. l'his is a tautology because the push forward has already been defined as the dual to the usual pull back.

Operations with cunents

93

many currents can be pulled back in the Since 7Y(1l') is dense in same way that differential forms are pulled back. For example, suppose

T= lIl=r

where each T1 is a locally mtegrable function on 1?'. Then

FT= Another example of pull back is given in the following lemma, which we

will need for Part 1V. LEMMA 6

x RN RN be given by Let i-: RN x RN; x E RN } (the diagonal of

r(y,x) = x — y. Let = {(x,x) E x Rh'). Then r [0] = [s]. An appealing formal proof can be given by writing [0] and as

PROOF

forms with distribution coefficients. From the examples early in this chapter, we have [0]

=öo(x)dxi A...AdXN

=

— y)d(xi

—

y') A... A d(xN — YN).

The lemma is then formally proved by replacing x by r(y, x) by dr3(y,x) = d(x, — y3).

=

x — y and

The above argument can be made rigorous by approximating [01 by a sequence

of smooth forms. However, here is another approach using Definition 2. Let We write E VN(RN x

III+IJI=N

where

each Øjj

is

an element of V(1W' x RN). Then

(r[01, Ø)RN xRN = ([0], TSØ)RN

= ([0],

J

where

III+IJI=N

(by Definition 2) (by Lemma 1)

Currents

94

only the piece of integral, we obtain Since

f

x + y) of degree N

in

dy

contributes

to the above

f

III+IJI=NVERN

Therefore

f

III+IJI=NVERN

=

XERN

J

111+1 JI=NVERN

= as

desired.

I

Finally, we mention the analogues of Lemmas 3 and 4 for pull backs. We leave the easy proof to the reader. LEMMA 7

and Il' are open subsets of oriented manifolds of dimensions N and Il' is a smooth surjective map such that —' has maximal rank = N' for each p eft Then F* o d = do F as operators from D"(fZ') to (1Z). If in addition and Il' are complex manifolds and F is holomorphic, then F* o 8 = Oo F* as operators from to Suppose

N' (N N'), respectively. Suppose F:

Part II CR Manifolds

In the first part of this book, we defined two types of manifolds: the smooth manifold and the complex manifold. Real analytic manifolds were also defined, but aside from the real analyticity of the coordinate functions, the structure of this class of manifolds differs little from the structure of the class of smooth manifolds. Complex manifolds are fundamentally different because of their additional structure. A complex manifold gives rise to a complex structure map (J) and a Cauchy—Riemann operator (0) which are not part of the structure of a smooth or real analytic manifold. In Part II of this book, a third class of manifolds is introduced — the CR manifolds. The CR stands for Cauchy—Riemann and as the name suggests, a CR manifold retains some of the additional complex structure of a complex manifold. Indeed, any complex manifold is a CR manifold. However, the class of CR manifolds is much more general. It includes the class of all real hypersurfaces in In fact, "most" submanifolds of are CR manifolds. The basic definition of a CR manifold is given in Chapter 7. Both abstract and imbedded CR manifolds are discussed. In Chapter 8, the tangential Cauchy—

Riemann complex is introduced. This is analogous to the 0-complex for a complex manifold. Here, two points of view are discussed. For a CR submanifold of C's, an extrinsic point of view is given which relates the tangential Cauchy—Riemann complex to the 0-complex on the ambient For an abstract CR manifold, an intrinsic point of view is presented which requires no ambiently defined Cauchy—Riemann complex. These two points of view are then shown to be isomorphic in the case of a CR submanifold of In Chapter 9, the concept of a CR function is introduced. A CR function on a CR manifold is analogous to a holomorphic function on a complex manifold. However, the behavior of a CR function can be much different. For example, a CR function is not always smooth or even continuous. Holomorphic functions on always restrict to CR functions on a CR submanifold. However, a CR function on a CR submanifold does not always extend to a holomorphic function on some open

95

96

CR

subset of C's. The question of the holomorphic extension of CR functions will be taken up in detail in Part III of this book. In Chapter 10, we introduce the Levi form, which is analogous to the second fundamental form in differential geometry. The Levi form is the key differential geometric object which determines many function theoretic properties of a CR manifold (for example, the holomorphic extendability of CR functions and the local solvability of the tangential Cauchy—Riemann complex). In Chapter 11, we discuss the imbeddability of CR manifolds. We show that a real analytic CR manifold can always be imbedded as a CR submanifold of The C°° version of this theorem does not hold. Nirenberg's example of a three-dimensional COC strictly pseudoconvex manifold that cannot be imbedded in any is presented at the end of Chapter 11.

7 CR Manifolds

We start this chapter with the definition of an imbedded CR manifold, which is the simplest class of CR manifolds. Later, we will present the definition of an abstract CR manifold.

7.1

Imbedded CR manifolds

Here, we will concentrate on the case of a CR manifold imbedded in Cs'. We could easily replace by a general complex manifold but this would unnecessarily complicate matters and add little to the understanding of the subject of CR manifolds. For a smooth submanifold M of recall that is the real tangent is not invariant under the space of M at a point p E M. In general, complex structure map J for Therefore, we give special designation to the largest J-invariant subspace of DEFINITION 1

For a point p E M, the complex tangent space of M at p is

the vector space

=

fl

(M) is sometimes called the holomorphic tangent space. This The space must be an even-dimensional real vector space because

Jo JIHP(M) = and therefore

(detJIH9(M)]2 =

where m = dimR Note that if A : (i.e., J o A = A o J), then C

'—'

is a complex linear map

97

CR Mantfolds

98

which

We also give special designation for the "other directions" in do not lie in DEFINITION 2

The totally real part of the tangent space of M is the quotient

space

= with we can identify Using the Euclidean inner product on With this in (denoted the orthogonal complement of is the fl X,(M) = {O}, because identification, note that = We have e largest J-invariant subspace of Therefore, is orthogonal to From Lemma 1 in Section 3.2, is transverse to are of crucial importance. and The dimensions of LEMMA 1

of

Suppose M is a real

of real dimension 2n —

d.

Then

and

PROOF

C T,,(M) and so

First note that

= 2n — d.


dimR

To establish the other inequality, note

j

+

and so + dimR

J from dimR

dimR

dimR

=

dimR

the statement of the dimension of

dimft

—

in the lemma also follows.

I

is called the CR codimension of M. is an even number between 2n — 2d and and so the only possibility is 2n — d. If M is a real hypersurface, then d = never changes. = 2n — 2. In particular, the dimension of dimR If d> 1, then there are more possibilities. Consider the following example. The real dimension of

The lemma states that dimft

1

Imbedded CR mantfoids

99

Example 1 Let M = {z E C'2;IzI = 1 and Im z1 = O}. M is just the equator of the unit sphere in C'2. Here, d = 2 and so 2n —4 < <2n —2, for (zi z2 = 1, , z,, (M) p E M. At the point p' = z3 = 0, = 0, = 0) E M, , is spanned over R by {O/Oxi, O/0y2, 0/Ox3, }. The vectors J(8/Oxj) = O/Oyi and J(O/8y2) = —(O/8x2) are orthogonal to (M) and therefore 0/Ox,, O/8y2 span (M). The vectors {O/0x3, span the J-invariant subspace (M). So in this case, O/Oy,, } (M) = 2n — 4 and dimR X,, (M) = 2. dima = (z1 = 1, = 0, . . . , z,, = 0) E M. Here, Now consider the point which is is spanned (over R) by {O/8x2,O/OTh,... = and X,,2(M) = {O}. In this case, J-invariant. Therefore, U

In the above example, the dimension of

requirement of a CR manifold is that dime

vanes with p. The basic is independent of p E M.

A M of C" is called an imbedded CR is independent of p E M. or a CR submantfold of C" if dimR

DEFINITION 3

Any real hypersurface in C" is a CR submanifold of C". Another class of CR submanifolds is the class of complex submanifolds of C". For a complex submanifold M, the real tangent space is already J-invariant and so = Another example of a CR submanifold is a totally real submanifold, which is on the opposite end of the spectrum from a complex manifold.

DEFINITION 4 A submanifold M in C" is said to be totally real if {0},for each p E M.

=

= for p E M. From Lemma 1, the real dimension of a totally real

An equivalent definition of a totally real submanifold is that

submanifold is at most n. An example of a totally real submanifold is the copy 0}. Any smooth graph over this copy of R" is also a totally real submanifold. The complexifications of and are denoted by T,(M)® ® C, respectively. The complex structure map J C, 0 C, and on ® C restricts to a complex structure map on 0 C because

of R" given by {(x + iy) E C";y

is J-invanant. From Section 3.2,

® C is the direct sum of

the +i and —i eigenspaces of J which are denoted by

and

CR Manifolds

100

respectively. We have

n

=

® C}

=

C}

It will be useful to have a way of identifying the above spaces in terms of a local defining system for M. We have the following lemma. LEMMA 2

defined near a point p E M by M = Let M be a smooth submanjfold of { z E C";pi(z) = pd(z) = O}, where p1,... ,Pd are smooth real-valued functions with dpi A... A dpd 0 near p. (a)

A vector W =

w3

if and only

belongs to

E

If W{pk}(p) =

(Opk,

=

1 k d.

= 0, 2

(b)

A

vector W =

w2

belongs to Hg" (M) if and only

if

that (

,

PROOF

W{pk } denotes the action of a vector W on a function Pk. and ),, denotes the pairing between forms and vectors.

We have

=

fl {T,(M) ® C} and for

1

because 0pk is a form of bidegree WE Clearly, (Opk, if and only if (0, 1). In addition, dpk = (9f)k + OPk. Therefore, W E (Opk, W)p = 0. Part (b) is proved the same way. I

Imbedded CR

101

If M is a CR submanifold of

then the dimensions of 0 C are independent of the point p E M. We define the following

and

subsets of Tc(M):

HC(M)=

U

pE M

H"°(M) = U

pE M

H°"(M) = U

pE M

As mentioned in Part I, a subbundle of Tc (M) is an object that assigns to each

point p E M a subspace of 0 C whose dimension is independent of p. In addition, these subspaces are required to fit together smoothly in the sense that they are locally generated by a basis of smooth vector fields. l'his latter requirement is easily seen to be satisfied by the spaces H"°(M), H°"(M), and Hc(M). For if M is locally defined by then near a point P0 E M, . we can choose from {OPi,. . '9pd} a collection of k-elements 8p11,. .. .

k

(1

,

d) that axe linearly independent. The number k is the CR codimension

of M. From elementary linear algebra, there are smooth linearly independent vector fields L1,. . . , that are annihilated by These vector fields locally generate H"°(M) by Lemma 2, and so H"°(M) is a subbundle . . .

,

of Tc(M). Similarly, H°"(M) and Hc(M) are also subbundles of Tc(M). In the next section, we will construct local bases for H"°(M) and H°" (M) in a canonical way after locally graphing M over its tangent space. LEMMA 3

Suppose M is a CR (a) (b)

of

Then

{O} for each p EM. The subbundles H°" (M) and H"°(M) are involutive.

Recall that a subbundle is involutive if it is closed under the Lie bracket. PROOF

The proof of part (a) follows from the fact that the intersection of

eigenspaces of any linear map corresponding to different eigenvalues is always trivial. For part (b), we first note

H"°(M) = TC(M) The bundle T"°

fl

is involutive because the Lie bracket of any two vector fields spanned by O/Ozi, . . . , is again spanned by {O/Ozi,... , }. In addition, Tc (M) is involutive because the tangent bundle of any manifold is involutive. So, H1'°(M) is involutive, as desired. Since H°"(M) = H"°(M), 11°" (M) is also involutive. I

CR Man(folds

102

The above lemma is important because properties (a) and (b) are the defining properties of an abstract CR manifold (see Section 7.4). The lemma does not

(M) = H"°(M) 11°" (M) is involutive. In general, this is not imply that true. In fact the Levi form, discussed in Chapter 10, measures the degree to which HC (M) fails to be involutive. Lemma 2 implies that dimc H"°(M) = n — k where k is the number of linearly independent elements of (p),.. . , Opd(p)}. If Opi A ... A '9Pd 0 = n—d = dimc and so dimR = 2n—2d. then dimc According to Lemma 1, this is the minimum value of the dimension for The condition 0P1 A. . . A 0 is an open condition. It is also generic in the sense that a random collection of defining functions will, with high probability, satisfy this condition. DEFINITiON 5 A CR submanifold M is called generic if dimR

is min-

imal.

A real hypersurface in is always generic. By Lemma 1, a generic CR submanifold of whose real codiniension is at least n must be totally real. Any complex submanifold of that is not an open subset of C'2 is an example of a nongeneric CR submanifold. The following lemma follows easily from the definitions and Lemma 2. LEMMA 4

Suppose M is a CR submanifold of C'2 with dimR M = 2n The following are equivalent.

—

d,

0 d ii.

(a) M is generic. (b) (c)

(d)

= 2n — 2d, for p

M. The CR codimension of M equals d. A. . . A ôPd 0 on Mfor each local defining system {p',. .. , pd} for M. dimR

(e)

=

Let us return to the above example of the equator in the unit sphere, M = = I and Im zi = 0}. In this case, M is the common zero set of We have the defining functions p1(z) = 1z12 — 1 and p2(z) = (1/2i)(zi — { z E C'2; IzI

ôp1(z) = 2•dz = 0p2(z) Clearly,

=

A normal form for a generic CR submanifold

=

0,

z1 = ± 1. So, M is

103

not a CR submanifold. However, the set

M= {Z =

E M;z1

±1}

is a generic (noncompact) CR submanifold of

hypersurface, then is a one (real) dimensional subspace of By Lemma 1 in Section 3.2, is orthogonal to both and Therefore, can be identified with the orthogonal complement of For higher codimension, both and are orthogonal to (by Lemma 1 in Section 3.2). However, in general, is not orthogonal to and therefore cannot be identified with the orthogonal complement of in any natural If M is

way.

a real

Consider the following example: let

M={(Zi,Z2)EC2; Imz1 =Rez2,

M is

totally

real and a basis

for To(M)

18 iOXi

= O/Oyj is

is given

ImZ2

=0}.

by

8 8 —+— Ox2 Oyj

Note

that J(8/Oxi)

7.2

A normal form for a generic CR submanifold

not orthogonal

to O/Oy' + 0/Ox2.

In this section, we present a convenient coordinate description of a generic CR

submanifold. As a start, we locally graph the CR submanifold over its real tangent space. Then, we show that the graphing function can be chosen so that certain "pure terms" in its Taylor expansion vanish. The real analytic case is handled first. From this, a version will easily follow. We start with an easy warm-up. LEMMA

I

Suppose M is a smooth generic CR submanifold of Ctm with dimR M = 2n — d, 1 d n. Suppose p0 is a point in M. There is a nonsingular, complex affine linear map A: an open neighborhood U of and a smooth —p function h : R" x Cn_d Rd with h(0) = 0 and Dh(0) = 0 such that

A{MflU}={(x+iy,w)EC" xCtm_d;y=h(x,w)}. First, we translate so that gular complex linear map A:

is the origin. It suffices to find a nonsinthat takes To(M) to the space {y = 0} in where y = Im Z E Rd. For then the graphing function, h, for A{M} will satisfy the required properties. PROOF

CR Manifolds

104

To

find the desired map A, let

. .

,

be an orthononnal basis for Xo(M).

The J-invariant space Ho(M) is orthogonal to both Xo(M) and J{Xo(M)}.

In addition, J has no real eigenvalues and Jv w = —v Jw for v, w E Therefore, there is an orthononnal basis for Ho(M) of the form Since M is genetic, the set Vd+ 1,Ji)d-4-1,.. , .

{i)1,Ji)1,. .. is

a basis for

by Lemma 4.

Nowletz=x+iyECd

Here,xandybelongto

Rd and u, v belong to Rn_d. We define the real linear map A: its action on basis vectors 8

8

A(I,)=

1

by

< — j —

d+l

ÔVd

J(O/8x2) O/8y3 and J(t9/Ouk) = t9/t9vk, clearly AoJ = JoA on the basis for and therefore AoJ = JoA on all of Viewed as a map from to C's, A is complex linear. Clearly A{To(M)} = {(x, 0, u, v); x E Since

{y = 0}, as desired. This completes the proof of the

and u, v lemma.

I

REMARK

Since A

is

so the extension of A

complex linear, we have to

= Ho(AtM}) and = (A{M})

0 C satisfies

A{H°"(M)} =

In addition, A sends an orthonormal basis to an orthonormal basis for To(A{M}). In particular, Xo(A{M}). In the new coordinates — relabeling A{M} by M — we have and

for

To(M)={(x,0,u,v);xeRd

and

Ho(M) = {(0,O,u,v);u,v e Xo(M) = {(x,0,0,0);x

E

can be identified with {(0, w); w

in

a complex linear fashion,

where w = u + iv.

Most of our concern in this book will be with generic CR submanifolds. However, we should point out that a different version of Lemma 1 holds in the nongeneric case. Let k be the CR codimension of M and let d be the real codimension of M. For the nongenenc case, we have 0 k < d and so n dirnc H"°(M) > n — d by Lemma 1 in Section 7.1. Define the integer i

105

A normal form for a generic CR

by dim 14'°(M) = n — d + j. We have

= dimc H'°(M) + dimc H°'(M) + dima X(M)

2n — d

=2n—2d+2j+k. Therefore, 2j + k =

d.

is a k-dimensional subspace that Since of real there is a J-invariant subspace of

is transverse to dimension 2j that is transverse to

}. The proof of Lemma 1 can be modified so that after a complex linear change of coordinates, the defining

equations for M become

=

• ,Xd_j,WI,... ,Wn_d+j)

= Yj+1

Yd—j = hd_,(x3+I,. .. ,Xd_,,W1,..

.

are smooth complex-valued functions with Ht (0) = 0 and are smooth real-valued , DHt(0) = 0, 1 £ j, and where functions with ht(0) = 0 and Dhe(0) = 0 for j + 1 £ d — i. Now we present a normal form for a real analytic, CR, generic submanifold. This is the first step toward a more complicated normal form of Bloom and Graham [BG], which we will discuss in Chapter 12. where H1, ..

.

,

H3

THEOREM 1

with dimR M = Suppose M is a real analytic, generic CR submanjfold of 2n — d (1 d n). Suppose that p0 is a point in M. There is a neighborhood C Ctz; and a real analytic U of p0 in h: Rd x Cnd with h(0) = 0 and Dh(0) = 0 such that :

fl U}

c

= {(x + iy, w) E

Cd x

Furthermore

(0)=0 h

for all multiindices a and /3.

(0)=0

y = h(x, w)}.

CR Manifolds

106

Another way to describe the graphing function h is to look at its Taylor expansion about the origin

h(x,w) =

a

a!/3!'y!

- (0). 0w1

The terms in the Taylor expansion with either = 0 or 'y = 0 are called pure terms. The content of the theorem is that with the proper choice of holomorphic coordinates, the graphing function for M has no pure terms in its Taylor expansion. PROOF

From Lemma 1, we may assume that the given point

is the origin

and

where h(O) = 0 and Dh(0) = 0. Since M is real analytic, its graphing function h: Rd x Cn_d is real analytic. The Taylor expansion of h is given by

h(x,w,zD) =

Note we have emphasized that h is not holomorphic in w by the notation h(x, w, tD), which illustrates the dependence of h on zD. We replace iZ' by an independent coordinate holomorphic map h: Cd x

and x by z E Cd

E

Cnd

x

and

we define the

Cd by

h(z,w,i7) = This series converges for {IzI, wi,

h(0) =

and Dh(0) =

< 6} for some 6 > 0. we have h(0) = 0 and Dh(0) =

0. By the implicit function theorem for holomorphic maps, there is a unique holomorphic map Cd x Cn_d —p Cd defined near the origin such that

Since

0

0,

+ i&(z, w, 0), w) =

z

for

zI, wi < 6

(1)

for some, possibly smaller 6 > 0. Define the holomorphic change of variables (2, t1) = 4(z, w) where

2=

z —

w), w, 0) E Cd (2)

Since Dh(0, 0,0) =

0,

D4(O, 0) is the identity. So for an appropriate 6 > 0,

the map ci, is a biholomorphism from {izl, Iwl < 6} to a neighborhood of the origin in C'2.

A normal form for a generic CR submanifold

107

Let M be the image of M fl {IzI, wi < 6} under 4. We wish to find a defining function for M in the 2 = ± +

th coordinates of the form

so that h satisfies the required properties stated in the theorem. From (2),

=

y

—

If (2, ti)) belongs to M, then (z, w) belongs to M and so

y=

E

Rd.

Substituting this equation into the previous one, we have

= Re{h(x, w, ii') —

+

w, iD), w), w, 0)}

(3)

for (2,th) = 4'(z,w) E M. To obtain a defining equation for M in the (2, th) coordinates, we must transform the right side of (3) into a function of ± and ti). Now since Z is a local biholomorphism, clearly (z, w) = z

4''(2, th) and we write

= z(2,ti')

WW where z: Cd x Cn_d

Cd is holomorphic near the origin. Therefore x = Re z is a real analytic function of Re 2, Im 2, Re th, Im th and we write x = x(2, th). Substituting x = x(2, ti') andw = tl, into the right side of (3) would result in a local graphing function for M, except that this would still involve the variable = Im 2. A graphing_function should only involve ± and th. To remedy this, we use the fact that Dh(0, 0,0) = 0 and the implicit function theorem to solve for the variable = ti,, in (3) (with x = x(2, ti)) and w = ti)) as a real analytic function of ±, Re ti', Im th near the origin. Substituting t1, ti)) for = Im 2 in x(2, yields a new function which we denote by x(±, tl,, ti,). This function is real analytic in a neighborhood of the origin. Now substituting x(±, ti), for x and ti) for w, (3) becomes

=

for

(2,th) E M

where

= —h(q5(x(±, th,

+

iii(x(±, a,,, ii,,, th), ti), 0).

Therefore, Re h is the graphing function for M.

CR Manifolds

108

It remains to show

Re h

=0

(0)

Re /i

and

(0)

(4)

0.

The second equation follows by conjugating the first, so it suffices to show the first.

The function

(1, th) '—i x(1, th,,) real analytic near the origin and therefore can be expressed as a power series in i, th, and th. Replacing by i Cd and ti) by we obtain a map x: C' x Cn_d x -d _, Cd is

which is holomorphic in a neighborhood of the origin. By substituting

th,

for x(i, th, th) and for th in the definition of h, we obtain a function h: C" x x Cn_d Cd given by

= th,

+

i)), th), ti,,0)

th,

which is holomorphic for (1, th, in a neighborhood of the origin. To establish the first equation in (4), we must show

=0. From (1) with z replaced by x(1,i1,0) and w replaced by ii', we obtain

0) +

0),

0), t1) = x(1, ti,, 0).

Substituting this into h gives 0), th, 0)

th, 0)

as desired. Therefore (4) holds and the proof of the theorem is complete.

I

A C" version of Theorem 1 follows easily from the real analytic version.

Suppose M = {y = h(x, w)} where h is of class C" for k 2. From a kth order Taylor expansion of h, we have

h(x, w) = p(x, w, th) + e(x, w)

where p is a polynomial of degree k in the variables x, w,

and where the

Taylor remainder e satisfies e(0)

=0

for

+

+

k.

A normal form for a generic CR submanzfold

109

the manifolds M and M = {y = p(x, w, to order k at the origin. Since p is real analytic, Theorem 1 applies to Al. The images of M and M in the new coordinates still agree to order k at the origin. We Therefore,

have established the following. THEOREM 2

(k 2) with Suppose M is a generic, CR of C'1 of class dimR M = 2n — d (I d n). Suppose that p0 is a point in M. There is a neighborhood U of po in C'1; a biholomorphism c1: U C C'1; and a function h: Rd x Cn_d Rd of class Ck with h(0) = 0 and Dh(0) = 0 such that

c

n U} = {(x + iy, w)

Cd x

h(x, w)}

Furthermore 0

0

Ox°Ow Now we turn to the question of finding a canonical local basis for H"°(M) and H°" (M). We assume that Al is graphed over the origin as in the conclusion of Lemma 1, i.e.,

M = {y = h(x,w)} x Cn_d

Rd is of class CC, fork 2, and where h(0) =0 and Dh(0) = 0. The sjace (M) is spanned (over C) by 8/thD1, . and the space (M) is spanned by 8/Ow,, . , It is our desire to extend these vectors to vector fields that locally generate H°"(M) and H"°(M), where Ii:

. . ,

. .

respectively. THEOREM 3

pdxCn_d RI! is of class Ck (k 2) with h(0) = 0 and Dh(0) = 0. A basis for H'°(M) near the origin is given by L1,... , L3 = where

is

with

i—

1

the (€, k)th element of the d x d matrix

/ I .Oh\' \ Ox

I

—

A basis for H°'(M) near the origin is given by

n —d

CR

110

Let

PROOF

1j

= where

for 1 j

=

Since Dh(O)=O, note that LjIO=:O/OWj and

the Ak3 are smooth functions to be chosen so that L3 IM

belongs

to

H"°(M). Let p3(z,w) = Im

The manifold M is the — h,(x,w), 1 j d. common zero set of Pt,... , Pd. By Lemma 2 in Section 7.1, a vector field L in belongs

to H"°(M),

provided

on M for 1 £ d,

L) = 0

where ( , ) denotes the pairing between 1-forms and vectors. Inserting L = L, into this equation, we obtain 1

Thiscanberewriueninmatrixformas

1

11 formula for the L, H"°(M) the set

The

given

.Ohl

Oh

in the theorem now follows. Since H°" (M) . .

forms

a local basis for H°"(M).

I

If the graphing function h for M is independent of the variable x, then the local basis in Theorem 3 for H"° (M) has the following simpler form: L3 =

0 Ow3

d

-

0

1 3 n—d.

Ow, Ozi

We give such a manifold a special name. DEFINITION 1

A CR submanifold of the form

M = {(x + iy, w) E where

h:

CTL_d

:

Rd

= h(w)}

x

is smooth with h(0) =

0

and Dh(0) = 0 is called rigid.

In Chapter 12, we will give an intrinsic, coordinate-free description of rigidity due to Baouendi, Rothschild, and Treves.

Quadric submanjfolds

7.3

111

Quadric submanifolds

By Theorem 2 in the previous section, a smooth, generic, CR submanifold of has a locally defined graphing function h with no pure terms in its Taylor expansion up to a given order. In particular, the quadratic terms in the Taylor expansion of h contain no terms involving the x-coordinates. Therefore all the second-order information is contained in the term -

n—d

q(w,w)=

-

0h2(O)

wjWk. k

By replacing ti' by the independent variable valued) quadratic form.

DEFINITION I (i) (ii)

A map q:

x

E Cn_d, we obtain a (vector

Cd is a quadratic form if

q is bilinear over C q is symmetric, i.e., q(w,ij) = q(17,w) for

e Ctm

(ill) DEFINITION 2 A submantfold M C C'2 defined by

M = {(x + iy, w) E Cd x

= q(w, t1)}

where q: Cn_d x Cn_d —+ Cd is a quadratic form is called a quadric subman-

ifold of C'2.

Requirements (ii) and (iii) of Definition limply that q(w, tii) is a vector in Rd. Hence, the quadric submanifold in Definition 2 is a well-defined submanifold of real dimension 2n — d. Quadric submanifolds are rigid since their graphing functions are independent

of x. As we will see, the class of quadric submanifolds provides a class of easily studied examples. From the discussion at the beginning of this section, any generic CR submanifold can be approximated to third order at the origin by a quadric submanifold. Therefore, a quadric submanifold often serves as the model for a more general CR submanifold. Another reason why quadric submanifolds are interesting is that each quadric submanifold has a group structure, which we now describe. DEFINITION 3 Let q: (n_d x (z2,W2) E Cd x Cnd define

—+ C'

be a quadratic form. For (z1, w1),

(z1,wj)o(z2,w2)=(zj+z2+2iq(wi,iD2),wi +w2).

CR Manifolds

112

LEMMA I

The operation o defines a group structure on C'2 x C'2 that restricts to a group

structure on M x M, where M = {y = q(w, The operation o is easily shown to be associative. The identity element is the origin. The inverse of the point (z, w) is the point PROOF

(z,w)' = (—z+2iq(w,ti),—w). It remains to show that o restricts to a group structure on M x M. If (z1, wi) and (z2, w2) belong to M, then Im z1 q(wj, and Im Z2 q(w2, W2). Therefore

+ q(w2,ti'2) + 2 Re q(wl,th2).

Im{zi + z2 + 2iq(w1,ti)2)} =

Using the properties of q from Definition 1, this can be rewritten as

Im{zi +z2+2zq(wi,w2)}=q(wi +w2,ti)i +W2). This shows that M x M is closed under o• We leave it to the reader to show that if (z, w) belongs to M then (z, w)' also belongs to M. The proof of the lemma is now complete. I The lemma implies that a quadric submanifold is a Lie group, which means

M is a smooth function. For M

that the group operation (o): M x M M (Z0, w0) E M, define the map (z,w) o

=

(zO,wO)

= (z+zo+2iq(w,'tho),w+wo). is a smooth function in both (z,w) and p0. For fixed p0, is the restriction of a holomorphic map since q is complex linear as a function of the first factor.

From Theorem 3 in Section 7.2, the generators for H"° (M) are

where we have written q = Likewise, the generators for H°" (M) , These vector fields are globally defined since M is globally , presented as the graph of q. Also note that 0/Oxi,... , O/Oxd are the global generators for the totally real tangent bundle X (M). are L1, .

. .

The vector fields L1,...

,

L1,..

. ,

and 0/Oxi,

.

.. ,

have

another important property. They are invariant under the group action (o) for w) sends the M. Let us explain what this means. The map (z, w) '—' is a complex origin, 0, to the point p0 = (zn, wo) E M. Therefore,

Quad& submantfolds

113

linear map from T0(M)ØC to and 3.1).

as explained in Part I (see Sections 2.3

Tc (M) is said to be (left) invariant for the

DEFINITION 4 A vector field L group action (o) for M if

=

for each p E M

= (z,w) op.

where

Since

=

o

and

o

=

). o (Gm)., an invariant

vector field L also satisfies for

pi,p2€M.

THEOREM I The vector fields

L,

and are

invariant for the group action (o) defined on M = {y = q(w,

PROOF

If C:

is a smooth map, then

=

+ +

G,(O)

= we have written (= ((i,...

as the coordinates for C'2 and we have , written the component functions of C as C = (G1, . , In our case, the map = is holomorphic in (= (z,w) C'2, and so 0 = O(G9)/O(j. Using the formula for the L, and writing

where

. .

0/Ox3 = O/0z3

the proof of the theorem reduces to an easy calculation.

Example 1 One of the most important examples of a quadric is the Heisenberg group. This is the real hypersurface in C'2 defined by

M = {(z,w)

C x C'2'; Im z

= 1w12}.

CR

114

= w

Here, ii

w = (w1,.. . The group action is given by

=

= (m,... ,

(z',w')o(z2,w2) for (z',w1),(z2,w2)

E

and

+w2)

=(z1 +z2 +2iw'

Cx

The invariant generators for H"°(M)

a

are

given

by

a

Oz

The invariant vector fields L1,.. .

,

vector field 0/Ox (where x =

Re

generate H°'1 (M), and the invariant

z) generates X(M). The vector fields

have interesting nonsolvability properties which will be discussed L1,. , at the end of Part IV. [I We conclude this section on quadric surfaces by deriving a normal form for the quadric surfaces of real codimension two in C4. This normal form will be useful for exhibiting various types of CR extension phenomena in Part ifi. Let (z1, z2, w1, w2) be the coordinates for C4 and write z1 = + iy1 and x2 + A codimension two quadric in C4 has the form

M = [Yi = 11/2

= q2(w,w)

and are scalar-valued quadratic forms on C2. We say that q1 and are linearly independent over R if the restrictions of q1 and to the set E C2} are linearly independent over R in the usual sense. From the properties of a quadratic form given in Definition 1, it is easy to see that and are linearly independent over R if and only if and as functions from C2 x C2 C are linearly independent over C. where

THEOREM 2

Suppose M is a quadric codimension Iwo submanifold of C4 defined by

11/2

(a)

If q1 and are linearly dependent over R, then there is a nonsingular complexlinear change of coordinates in C4 so that in the new coordinates

M=

q1 (w,

I Y2

where

=0

is a scalar-valued quadratic form.

Quadric submanifolds

115

Q2 are linearly independent over R then there is a nonsingular complex linear change of coordinates in C4 so that in the new coordinates, M has one of the following forms: If Qi and

(b)

(i)

Iyi = = = I. Y2 = I yj =

1w112 2

I..

(ii)

(lii) M=<

I

1/2

1w112

= Im(wl1i'2)

In Chapter 10, we will relate the quadratic forms q1 and to the Levi form. Then we will have a more intrinsic way of deciding which of the above normal forms applies to a given quadric submanifold in C4. Given a scalar-valued quadratic form q: C2 x C2 —* C, there is a 2 x 2 matrix with complex entries that represents q. There are complex numbers A, B, C, D with

q(w,17) = Since

(w1

W2)

(A D\(ij1 B

c )

q(w,ti)) is real valued and since q is symmetric, A and C are both real

and B = D. That is, the matrix that represents q is Hermitian symmetric. Throughout the proof of this theorem, we identify q with its matrix. The proof of part (b) will show that if qi or is positive or negative definite,

then the normal form given in (i) can be arranged. The same is true if both q1 and Q2 have a vanishing eigenvalue. If one of and (12 has a vanishing eigenvalue and the other has eigenvalues of opposite sign, then the normal form have eigenvalues of opposite given in (ii) can be arranged. If both and sign, then the normal form given in (iii) can be arranged.

For part (a), if = Aq1 for some A E make the following nonsingular complex linear change of variables PROOF OF THEOREM 2

= = Z2

— AZ1

w1=wI W2 =

W2.

In the new coordinates

= Im

= Im z1 =

= qi(ti,,th)

for

(z,w) E M

then we

CR Man qolds

116

and 1/2

= Im z2 =

Im

Z2 — A

= q2(w,w)

—

Im z1

Aqi(w,th)

for

(z,w) E M

=0. Therefore, if M is the image of M under the map (z, w)

t1),

then M is

defined by

Y2 = 0.

This completes the proof of part (a). For part (b), we start by diagonalizing the matrix of This can be done by a unitary change of coordinates which affects only wi and w2. After rescaling, there are three cases to consider:

1. qj(w,iJ)) = jwi 2 + 2. q1(w,tD) = 1w112 — 3. qi (w, ti)) =

IW212 1w212

w1 2

(qi is positive definite) (qi has eigenvalues of opposite sign) has one vanishing eigenvalue).

Note that case 1 also includes the case where q1 is negative definite, for in this case, the change of variables = —z and zi) = w will make the resulting q1 positive definite. We let

= where

A1w112 + 2

+ C1w212

A and C are real and A is complex.

Case 1. (with b

is positive definite). We first look for complex numbers a and b 0) and a real number t so that q2(w,

+ tq1(w, ti)) = lawi + bw212.

Expanding this equation, we see that a, b, and t must satisfy

ab =

A

1a12 = A + t

1b12 =C+t.

If A = 0, then we proceed as follows. We have A > C or C > A; for if A = C and A = 0, then and are linearly dependent. By switching the roles of w1 and w2 if necessary, we assume C > A. Set a = 0. This forces

t=—A. This inturn forces Ib12=C—A>0. Sowemayletb=i/C—A. With these choices, the above equations are satisfied with a nonzero b.

Quadric submanlfolds

If

A

117

0, then we must choose nonzero a and b. From the above equations,

we have Al2 A+t=lal =1W=c+t 2

This can be rearranged into the following quadratic equation in t

+ (A + C)t + (AC — Al2) = 0. Its discriminant, (A + C)2 — 4(AC — 1A12) = (A — C)2 + 41A12, is positive because A 0. Therefore, the above quadratic equation has two distinct real roots. Let t be the larger root. Clearly t + A > 0 and I + C > 0 because

(t + A)(t + C) =

Let 9 be an argument of A. The above three

1A12 > 0.

equations are satisfied by this choice of t and

a = (I + b

= (t + C)'12 > 0.

With this choice of a, b, t, we have q, (w, 11) = lwi 12 + 1w2l2

lawi

Define the following linear change of variables:

z2=z2+tz1 w2=aw,+bw2.

£1=z1

This is a nonsingular linear map since b Im

0. We have

= Im = q,(w,iD) = wi + =

(if (z,w) EM)

1w212

+

—

= alth,12 + 2 Re(-yth1th2) + i31th212

where a and

are positive real numbers and 'y is a complex number. Similarly

Imz2= Imz2+tlmz, =

=law,+bw212 =

(if (z, w) E M)

q2(w, tip) + tq1 (w,

lti'212.

by(1)

CR Manifolds

118

ti)), then Therefore, if M is the image of M under the linear map (z, w) i—' with = where in the new coordinates, M = {Im =

= alti'112 + 2 =

+ /31 W2I2

After dropping the A, we obtain

Now we complete the square in

qi(w, ti)) = Ia"2w1 +

+ (/3 — F112a1)Iw2I2

(recall that a > 0). We make one further linear change of coordinates

i,

2

a

—

Z2—Z2

)Z2,

= a"2Wi + 'ya"2vJ2,

W2 =

W2.

This change of coordinates is nonsingular since a > 0. Again, let M be the image of M under the map (z, w) (1, t1). The defining equation for M in the new coordinates is given by — Im z2 =

which is the normal form given in (i) of part (b) in the theorem.

Case 2. (w, = wi 12 — 1w212 (qi has eigenvalues of opposite sign). In this case, we make the following nonsingular linear change of coordinates: Z1=Z1

In

Z2=Z2

the new coordinates, (the image of) M is given by (after dropping the A)

I bn z1 = z2 = AIw1I2 + 2 Re(AwitD2) + B1w212

forsomechoiceofA,BER,andAEC. Let

A=r+si and r,sER. After the change of variables = is given by (drop the A) {Im z1 =

z1,

=

— 2rz1,

=

w1,

Im z2 =

Re(with2)

=

A1w112

—28 Im(w1i12) + B1w212.

In matrix form, we have -

-

-

q2(w,w) = (wt,w2)

(A is

—i8\(Wi

B)

= w2, M where

Quadrk submanifoids

119

If the determinant AR —

82

is positive, then the matrix of

is positive or

negative definite and so this falls under Case 1 above with the roles of and reversed. This leads to the normal form given in (i) of part (b). So we assume AB —

s2

0.

We first show that we can force the coefficient of Iwi 2 in change of variables of the form

z1=21 w1=th1

to vanish by a

z2=22 w2=tZ)2+itiI'1

for an appropriate t E R to be chosen later. Such a change of variables preserves qi. We obtain q2(w, tD)

(A + 2st + Bt2)1t11 12 — 2(s + Bt)

+

BItu'212.

is a real root t to the quadratic equation A + 28t + Bt2 = 0 because its discriminant 4(s2 — AB) is nonnegative. With this choice of t, the coefficient of itiii 2 vanishes and so we may assume (after dropping the A) There

qj(w,th) = Re(wiiD2) q2(w, tii) =

+ /31w212

and 0 are real numbers. 0. After a rescale in If = 0 then must be nonzero, for otherwise the z2-variable, M is in the normal form given in (ii) of part (b) with the roles of Wt and w2 and the roles of z1 and z2 reversed. if 0, then we can force the coefficient of 1w212 to vanish by a change of variables of the form where

ZIZ2

Z2Z2

w1—w1+itw2

W2—W2

t is a real number. Again, any change of variables of this form preserves We have

where

q2(w,w) =

+

CR Man Ifoldc

120

0, we may let t =

Since

which forces the coefficient of ki)212 to

vanish. After a rescale in the z2-variable, M is now in the normal form given in (iii) of part (b).

= lwi 2 (qi has a vanishing eigenvalue). We let

Case 3.

q2(w,w) =

A1w112 + 2 Re(Aw1i12) + B1w212

where A and .8 are real and A is complex. We make the change of variables

z2=z2—Az1 W1 =W1

W2

In the new variables, M is defined by I Im

= Itl,i

Im

2

2

Re(Athith2) + B1th212

Let q2(w, ti') = 2 Re(Awiü)2) + B1w212. The matrix that represents

is

(0 0, then the determinant of this matrix is —1A12 which is negative. Hence, the eigenvalues of the matrix of have opposite sign and this falls under Case 2 If A

above with the roles of and reversed. If A = 0 then, after a rescale, M has the normal form given in (i) of part (b). The proof of the theorem is now complete.

7.4

I

Abstract CR manifolds

So far, we have been dealing with CR submanifolds of In this section, we define the concept of an abstract CR manifold which requires no mention of an or complex manifold. ambient Let M be an abstract C°° manifold. As defined in Part I, Tc (M) denotes the complexified tangent bundle whose fiber at each point p E M is ® C. then from Lemma 3 in Section 7.1, If M s a CR submanifold of

(i) H"°(M) fl H"°(M) = {0} (ii) HLO(M) and H°"(M) are involutive. These two properties make no mention of a complex structure on other than to define the space H"°(M). Therefore, we define an abstract CR manifold to (M) which satisfies the above be a manifold together with a subbundle of two properties.

Abstract CR man(foldc

121

Let M be a C°° manifold and suppose L is a subbundle of Tc (M). The pair (M, L) is called (an abstract) CR manifold or CR structure if DEFINITION I

(a)

L is involutive, that is, [L1, L2] belongs to L whenever L1, L2 E IL

It is clear from the above discussion that if M is a CR submanifold of — then the pair (M,L) with L = H"°(M) is a CR structure. By analogy with the imbedded case, we call dimc {Tc (M)/L L} the CR codimension of (M, L). There is a complex structure map J defined on the real subbundle which generates L so that the eigenspaces of the extension of J to L L are L (for the eigenvalue +i) and L (for the eigenvalue —i). This follows from Lemma 3 in Section 3.2. In Section 4.3, we said that a pair (M, L) is an almost complex structure if L L = {O}. L is a subbundle of Tc (M) with Thus, an involutive almost complex structure is an example of a CR structure. As mentioned in Part I, the Newlander—Nirenberg theorem [NN] states that a manifold with an involutive almost complex structure is a complex manifold. (by definition), Now since a complex manifold can be locally imbedded into this prompts the analogous question for CR manifolds: if (M, L) is an abstract M CR structure, then does there exist a locally defined diffeomorphism

= so that '1(M) is a CR submanifold of Ctm with This last requirement for 1 implies that the CR structure for M (namely L) is pushed forward to the CR structure for 4{M} (namely H"°('I{M}). The answer to this question is a qualified yes. If M is real analytic, then there is a real analytic imbedding, as we will show in Section 11.1. If M is only smooth, then the answer, in general, is no, as we will show in Section 11.2, where we present Nirenberg's counterexample. There are further conditions on a smooth CR structure that will guarantee a local imbedding and we will briefly discuss these in Chapter 12.

8 The_Tangential Cauchy—Riemann Complex

For a CR submanifold of C'2, there are two ways to define the tangential Cauchy—

Riemann complex and both approaches appear in the literature. The first way The secis an extrinsic approach that uses the 0-complex of the ambient and ond way is an intrinsic approach that makes no use of the ambient therefore generalizes to abstract CR manifolds. In this chapter, we present both approaches. In the imbedded case, these approaches lead to different tangential Cauchy—Riemann complexes but in Section 8.3, we show they are isomorphic.

8.1

Extrinsic approach

Here, we assume the reader is familiar with the bundle of (p, q)-forms on C'2, over an The space of smooth sections of denoted Basic facts about the bundle of (p, q)open set U in C'2 is denoted forms and the associated Cauchy—Riemann complex 0:

—'

are

given

in Section 3.3. As also mentioned in Chapter 3, for each point p0 E C'2, the (C'2) is defined by declaring that the set Hermitian inner product on {dz' A = p. IJI = q, I, J increasing} is an orthonormal basis. Let M be a smooth, generic, CR submanifold of C'2 with real dimension to be the restriction of the bundle

2n—d. We define

is the union of where p0 ranges over to M, that is, M. This space is different from the space E AP'QT(C'2)}, where is an j: M —+ C'2 is the inclusion map. A smooth section of element of the form

f= 111p JI—q

whose

122

coefficient functions, fjj, have been restricted to M.

Extrinsic approach

123

For 0 p, q n, define the ideal in jp,q =

which is generated

by p and Op where p:

R is

any smooth function that vanishes on M Elements of jp,q

+

If {p',.

. . ,

are

sums of fonns of the type

A Op,

E

E

pd} is a local defining system for M, then {Pi, . .. , Pd} locally gen-

erates the ideal of all real-valued functions that vanish on M as shown in Lemma 3 of Section 2.2. Therefore, jp,q is the ideal in that is locally generated by

P1,...,Pd,OPI,••.,OPd. The restriction of jp,q to M, denoted is the ideal in locally generated by Op', .. , '9Pd. Since M is CR, the dimension of the fiber is independent of the point pij E M. Thus, is a subbundle of

Let

= I

the

orthogonal complement of jp,q I M in

L.

Elements in for E M are orthogonal to the ideal in (po)._ Let k be the number of linearly indepengenerated by Opi (P0),. .. , dent elements from (Po), . , (9pd(po)} (i.e., k is the CR codimension of M). Since M is CR, k is independent of the point p0 E M. Therefore, the dimension of is independent of the point p0 E M. Hence, the space is a subbundle of Note that =0 .

either p> n or q > n — k. If M is generic, then k = d (the real codimension of M) by Lemma 4 in Section 7.1. The space is not intrinsic to M, i.e., it is not a subspace of the exterior algebra generated by the complexified cotangent bundle of M. This is due to the fact that if p: R vanishes on M, then Op = (l/2)(dp+iJ*dp) is not orthogonal to the cotangent bundle of M due to the presence of J* dp.

For s 0, let =

..• e

where some of the summands on the right may vanish. The space is not the same as the space A8T*(M). The latter space is intrinsic to M whereas the former is not. As an example, let_M = {(z, w) E C2; Im z = 0}; then We have Op = p(z,w) = (2i)'(z — is and so the space of (p, q)-forms on M that are orthogonal to the ideal generated by

The Tangesnial Cauchy—Riemann Complex

124

In particular, A2.IT*(M) is generated by the form dz A dw A dii) and = A2.IT*(M), whereas A3T(M) is the 0. Therefore, space generated by dx A dw A dii) where x = Re z. Note that {jw;w E = A3T*(M). More will be said about f{AP.QT*(M)} in Section 8.3 AL2T*(M) =

where we discuss the relationship between the extrinsic and intrinsic tangential Cauchy—Riemann complexes.

For an open set U C M, the space of smooth sections of (M) over U will be denoted (U) will denote the space of compactly sup(U), and ported elements in If the open set U is not essential for the discussion, then it will be omitted from the notation.

For s 0, we let C8

—

where some of the summands on the right may vanish.

Again, note that

be the orthogonal projection map. For a form f E AP,qT* we often write ftM for 1M (f) and call this the tangential part of f. If / is a smooth (p, q)-form on then ftM is an element of Conversely, any form f E (U) can be extended to an where U is an open set in with Un M = U. This is element f E accomplished by writing Let tM:

fjjdz1 A

f=

with

fjj E E(U)

'I =p

JI=q and

extending each coefficient function fjj to an open subset U of

We now define the (extrinsic) tangential Cauchy—Riemann complex. DEFiNiTION I For an open set U C M, the tangential Cauchy—Riemann is defined as follows. For_f E complex OM: let with U fl M = U and let f E with ftM = / U be an open set in

on U fl M = U. Then OMf = The form OAif is calculated by extending / ambiently to an open set in Ci', then applying 0 and taking the tangential part of the result. Since there are many possible ambient extensions of a given element of we must show

that the definition of '9M is independent of the ambient extension. LEMMA

I

OM is well defined, that is, if fi and hare elements in

(f2)tM on MflU, then

= (0f2)tM on MnU.

with (f1)tM =

Extrinsic approach

125

Note that (f' — /2) is an element of Jp,q• l'herefore, it suffices to If show that 0 maps smooth sections of jp,q to and E U—+lRvanishesonMflU,then PROOF

=

+ fi A

+

+

A Op.

The iigbt side is clearly an element of

As already mentioned, the spaces (M) are not intrinsic to M. Thereand the resulting tangential Cauchy—Riemann operator are fore, the spaces

not intrinsic to M. For this reason, we refer to the above-defined tangential Cauchy—Riemann complex as being extrinsically defined. We shall specialize to the It is useful to have a procedure for computing case of a real hyperswface. LEMMA 2

p(z) =

Suppose M = {z

O}

is a real hypersuiface in

where

R is smooth with IdpI = 1 on M. Let N = 4(Op/Oz)

=

Then

= Nj(Op A

for

A

J is any ambiently defined smooth (p, q)-form with ftM = / on M. Recall that denotes the contraction operator of a vector with a form (see Section 1.5). The hypothesis that = I on M can easily be arranged by replacing p by p/IdpI. PROOF

Since Idp) =

1,

the vector field N is dual to the form Op. From

Section 1.5, if

and A t,t')

where (.) is the Hermitian inner product on

Therefore

=0 on M if and only if From the product rule for

(see Lemma 1 in Section 1.5), we have A

Since

=

q5 e

=

—

A

IdpI2 = 1 on M, this becomes (1)

The Tangential Cauchy-Riemann Complex

126

is an element of AP'QT(M). The ideal generated by p and lip).

0 and so Na(Op A form Op A (N1Ø) is an element of jp,q Now,

(the

Therefore, equation (1) provides an orthogonal decomposition of an element q5 of into its tangential part btM and the component of In particular, 4tM = A as claimed. The formula for and the definition of 8M• I OMf follows from the expression for The term NaØ is called the normal component of 0 and it is denoted by OflM.

Equation (1) then reads on M. 0= OtM + Op A This equation provides an orthogonal decomposition of into its tangential and normal components. If is an ambiently defined (p, q)-form on then both and are ambiently defined because p and hence N are ambiently defined. The equation = OtM + 0j A OflM also holds ambiently provided IdpI = I ambiently. This can be arranged, for example, if p is the signed distance function to M. ° At this point, the reader may wonder whether or not 00 = where j: M is the inclusion map. However, the right side of this equation does

not make sense because the domain of the tangential Cauchy—Riemann operator

which is not contained in A*T*(M), which is the range of j. The

is

right side of this equation does make sense if the tangential Cauchy—Riemann operator is defined intrinsically (see the next section). In Section 9.2, we shall

discuss CR maps (such as j) and the validity of commuting their pull backs with the tangential Cauchy—Riemann operator.

-

The following lemma follows easily from the analogous properties of 0 (see

Section 3.3). LEMMA 3

Suppose M is a smooth CR

of C's.

(a)

(b) OMOOM=O. From part (b), if OMf = g, then = 0. An important question is to ask whether or not the converse holds: if 0M9 = 0, then does there exist a form f with 8Mf = g? This solvability question for the tangential Cauchy—Riemann operator will be discussed in Part IV. It is useful to interpret the equation 0Mf = g in terms of currents (see Chapter 6 for basic facts about currents). If M is a smooth, oriented submanifold of of real dimension 2n — d (1
([MI,0)cn =Jo

for

0 E V2n_d(Cn).

Extrinsic approach

127

[M] is a current of degree d and therefore it splits up into its various bidegrees

[M] =

+

+

In particular E

where indicates the piece theorem, we have d[M] = 0. Since (0,d+ 1), we have

V2n_d(Cn)

of bidegree (n, n — d). By Stokes' is the piece of d[M] of bidegree o.

LEMMA 4

Let M be an oriented, generic, CR submanjfold of C'2 of real dimension 2n — d, 1 d n. Suppose f E Then ftM = 0 on M if and only if A f = 0 as a current on C'1.

PROOF By a partition of unity argument, it suffices to prove the lemma for forms f with support in a C'1-neighborhood U of a given point p0 E M. From A. . . A dpd where {pi,... , Pd} Section 6.1, the current [M] is given by is a local defining system for M near p0 and = Idpi A ... A dpdl'. Here, /1M is the distribution given by Hausdorif measure on M (see Section 5.1). The piece of bidegree (0, d) of this current is

A... A

=

A f = 0 if and only if

Thus for f E

OpIA...AÔpdAf=0 onMflU. Since M is generic, we have Op' A Op, A ...

A

0 and

Ôpd Af = OpiA ... A Ôpd A ftM

It follows that the equation Suppose

... A OPd

f is an element of

Af=

0

is equivalent to ftM = 0 on M.

Technically, the current

I

A f is not

well defined because [M]0.d is a current on C'2 and f is not ambiently defined on C'2. However, we can define A I as A f where f is any smooth, ambiently defined (p, q)-form with Lemma 4 implies that = f. this definition is independent of the extension f. With this in mind, we state and prove the following.

The Tangential Cauchy—Rlemann Complex

128

LEMMA 5

Suppose M is an oriented, CR, generic submanjfold of of real dimension 2n — d. Let f The equation bMf = g on M is and 9 equivalent to the current equation A [M]O.d) PROOF

From Lemma 4 and the fact that i9[M]O.d = 0, we have A [M]O.d) =

A

A

= Therefore, '9Mf = g if and only if Both

/E

C's.

on

gA

A [M)o.d) = g A

as desired.

I

the 0 and d operators satisfy an integration by parts formula. For we have

and g

(Of,g)cn = The same formula holds (by definition) if f is a current of bidegree (p, q). From

this equation, we will derive an integration by parts formula for

Recall that

the current pairing on M is denoted by (, ) M. We extend this current pairing to the spaces (which are not intrinsic to M) by setting

(f,g)M

=Jf A9 = ([M]Af,g)cn.

For a generic manifold M, and g for f = Oprovidedeitherp > norq > n—d. Therefore, iff

andg

en

(f,g)M = where and

is the piece of g of bidegree (n — p,n — q — d). If I E then f A 9 has bidegree (ii, n — d) and so

(f,g)M =11 Ag

=

A

LEMMA 6

Suppose M is an oriented, CR, generic 2n — d. Let f E and g E

=

of

with real dimension Then

Extrinsic approach

PROOF

129

From the discussion preceding the ststement of Lemma 6, we have

t

—

\VMJ , 9/M — \1LVJ J

where I E LP'Q(cn) of f and g. We have \(JMJ,9/M =

and

A g,'cn

,

are any ambient extensions

E

-'

,r,.xiO,d A

= = =

(by Lemma 4)

A Of,

(since

Af

0)

A

where the last equation follows from the definition of 0 applied to a current. = A Therefore By Lemma 4, we have [M]0.d A

= = as desired.

(—

IA

I

This integration by parts formula allows us to extend the definition of OM to currents on M. The space of currents on M of bidimension (p, q) is the dual of the space and it is denoted by }'. By adapting the proof of Lemma 1 in Section 6.1 to this context, the reader can easily show that

which is the space of currents of bidegree }' is isomorphic to n—q—d (n — p, n — q — d) on M. An element of is a form of bidegree (n — p, n — q — d) on M with distribution coefficients. {

Suppose M is an oriented, CR, generic with real dimension 2n — d. Let T E Dr', then aMT E

DEFINITION 2

of is the current

defined by

(OMT,g)M = From Lemma 3, we have

for g E

(T) = 0 for T E

In addition, Lemma 5

holds for currents, although we should say a word about the definition of [M]O*dA It suffices by a partition of unity argument 7 when 7' is an element of A Tlocally. Let p', . . . , Pd be a local defining system for M. to define is Hausdorif measure on M. A Then, A ... A 0Pd where = typical element of is T = T1 where T1 is a distribution on M and where with There is an element E 4e = on M. To define A T, it suffices to define the distribution 4UM . 7'1. By using a (smooth) x Cn_d; = 0}. local coordinate system, we may assume M = {(z, w) E In these coordinates, we have 4UM = where y = Imz. A distribution T1

The Tangential Cauchy—Riemann Complex

130

M acts in the variables x = Rez and w. Therefore, we define the distribution (ILM Ti)(x,y,w) = 6o(y) ® T1(x,w) where ® denotes the tensor product of distributions (see Section 5.2). The proof of Lemma 5 for currents now proceeds by approximating a current by a sequence of smooth forms and then using Lemma 5 for smooth forms. on

If M is not generic, then the CR codimension of M is less than d. In this case, the reader can show

= where

=

=

=

k is the CR codimension of M. Lemmas 4 through 6 hold with [M]0"

replaced by

8.2

Intrinsic approach to

Our treatment of the intrinsically defined tangential Cauchy—Riemann complex

is similar to that in [PWJ. We assume that (M, L) is an abstract CR structure (see Definition in Section 7.4). Therefore, L is an involutive subbundle of Tc (M) and L fl L = {O}. It will be necessary to choose a complementary subbundle to L L. In order to do this, we assume that M comes equipped with a Hermitian metric for Tc (M) so that L is orthogonal to L. If M is a submanifold of then the natural metric to use is the restriction to Tc(M) of the usual Hermitian metric on Tc (Ce). If M is an abstract manifold, then a metric can be constructed locally by declaring a local basis of vector fields to be orthonormal. This metric can be extended globally by a partition of unity. For each point p0 M, we let be the orthogonal complement of in C. Clearly, the spaces M} fit together smoothly (since p0 the do) and so the space 1

X(M)= U poEM

forms a subbundle of Tc (A'!)

—

then L = H"°(M) and L = H°"(M). In this case, X(M) is the totally real part of the tangent bundle.

If Mis a CR submanifold of Define the subbundles

T"°(M) = L

X(M).

We emphasize that T"°(M) is not analogous to H"°(M) for an imbedded CR manifold (unless X(M) = {O}).

Intrinsic approach to

The

131

dual of each of these spaces is denoted T°'3 (M) and T'°(M), respec-

lively. Forms in T*°" (M) annihilate vectors in T"°(M) and forms in T*'°(M) annihilate vectors in T°" (M) (by the definition of dual). Define the bundle

=

(M)).

This is called the space of (p, q)-forms on M. Unlike the extrinsic approach, these spaces are intrinsic to M (in the abstract case, they cannot be anything but intrinsic since there is no ambient space). Let m = dimc L and d =

dime X(M). Recall that d is the CR codimension of M. If p > m + d or q > m, then AP'QT*(M)

0.

The pointwise metric on Tc (M) induces a pointwise dual metric on T*C (M) in the usual way. Let be an orthonormal basis for and let

be an orthonormal basis for T°" (M). The metric for T* (M)

. . . ,

extends to a metric on A

Ill = p, IJI = q, I, J are increasing multiindices}

is an orthonormal basis. We also declare that M"?T*(M) is orthogonal to if either p r or q s. We have the following orthogonal decomposition ArT*C

(M) =

...

with the understanding that some of these suminands vanish if r > m. Let ArT*C

(M) —i AP.QT*(M) for p + q = r

be the natural projection map. The space of smooth r-forms on an open set U C M is denoted by and The space of smooth sections of AP'QT(M) over U is denoted by is the space of compactly supported elements of The "U" may be omitted from the notation if it is unimportant for the discussion at hand. The intrinsic definition of the tangential Cauchy—Riemann operator can now be given in terms of the exterior derivative dM: er'. DEFINITION 1 The tangential Cauchy—Riemann operator o is defined by 0M =

This definition of 8M is analogous to the definition of 8 on C" (or any other complex manifold). —p We will show that OM: is a complex, i.e., OM oOM = 0. This will follow from the equation dM o dM = 0 and type considerations. First, we need a preliminary result.

The Tangential Cauchy-Rlemann Complex

132

LEMMA I

If M is a CR

then

,,

icp,qi

çp+2,q—I

çp+i,q

Conceivably, the exterior derivative of a (p, q)-form, 0, might be a sum of forms of various bidegrees. The point is that the only possible nonvanishing components of dM0 have bidegrees (p+ 2,q — 1), (p+ 1,q) and (p,q + 1).

The case p =

PROOF

1,

q=

(dMO,iJi AL2) = 0 where

(,

0

will be handled first. We must show

This is equivalent to showing

7r°'2(dM(b) = 0 if 0

Et =

for all

) denotes the pairing between forms and vectors. From Lemma 3 in

Section 1.4 (dMq5,Tj AL2) —

Since

E

and L1,L2

(0, [Li, L.2]).

T°"(M), we have

0.

By the

definition of a CR structure, L is involutive, and so [Lj, L2] L. Therefore (0, (L1, L2]) 0, from which (dM0, L1 A L2) = 0 follows, as desired. This proves the lemma for the case p = 1,q 0. Note that the lemma automatically holds for p = 0 and q = I. For p, q 1, is generated by the following terms: OIA ... A

A

...A

is a smooth (1,0)-form and each is a smooth (0, 1)-form. The general case now follows by using the product rule for dM and the lemma for the case of a (1,0)-form or (0,1)-form. I where each

The

key ingredient of the proof is that T°" (M) = L is involutive. Since

T"°(M) is not necessarily involutive, we cannot conclude that 0

for 0 E

so

=

If M is a complex manifold, then T"°(M) is involutive and = 0 for 0 E This is a key difference between the

class of complex manifolds and the class of CR manifolds that are not complex manifolds. LEMMA 2

If M isa CR manifold, then ÔM o8M =0.

intrinsic approach (0

133

OM

is a smooth (p, q)-form; then ÔMØ =

PROOF Suppose Lemma 1 implies

=

dM0 —

(dM0).

+

Therefore

= =

(dM0) +

the term on the

dM

right vanishes. Therefore, OMÔMq5

=0, as desired.

I

The following product rule for OM follows from the product rule for the

exterior derivative. LEMMA 3

If f

and 9 E

then

OM(f Ag) = From Stokes' theorem and the product rule for 8M we obtain an integration

by parts formula for 0M• LEMMA 4

If I E

and g is any smooth form on M with compact support, then

=

Let m be the dimension of L (and L); let d be the dimension of X(M). Therefore, = m. dimc TLO(M) = m + d and dimc Tc (M) has complex dimension 2m + d which is the same as the real dimension of M. So a form of top degree on M has bidegree (m+d, m). 1ff then and so t9Mf pairs with forms of bidegree (m+d—p,m—q— 1). ÔMf From the product rule, we obtain Let g PROOF

Ag The

=JÔM(f Ag) +

A

8Mg.

(1)

bidegree of fAg is (m+d,m— 1). Since top degree on Mis (m+d,m),

we have

OM(f Ag)

=dM(fAg).

The Tangential Cauchy-Rlemann Complex

134

Since g has compact support on M, we have

fdM(f Ag) =0 by Stokes' theorem. This together with (1), establishes the lemma.

I

Just as with the extrinsic case, the integration by parts formula allows us to extend the definition of 0M to currents. By definition, the space of currents of bidimension (p, q) on an open set U C M is the dual of the space By adapting the proof of Lemma 1 in Section 6.1, the reader can easily show that this is isomorphic to the space of currents of bidegree (m + d — p, m — q) which is denoted by Elements in this space are (m+d—p,m—q)forms on U with coefficients in D'(U). If the open set U is not essential to the discussion, then it will be omitted from the notation.

DEFINITION 2

if T E

then the current OMT E

is defined by

(bMT,g)M = (_1)(T,9Mg)M. together with Lemma 2 shows that - An easy argument using this definition 'pq

OM(OMT)

8.3

=0 for a current T VM

The equivalence of the extrinsic and intrinsic tangential Cauchy—Riemann complexes

For a CR submanifold M of there is a choice of viewpoints for the tangential Cauchy—Riemann complex — the extrinsic and the intrinsic. These two complexes are different, but in this section, we show they are isomorphic. Before we establish this isomorphism, let us precisely define an isomorphism between two complexes. DEFINITION 1

(a) A complex is a collection of vector spaces A = {Aq; q E

o Z,q 0} with maps such that = Ofor q 0. (b) Suppose A = {Aq, dq; q 0} and A = {Aq, Dq; q 0} are two com-

plexes. These complexes are isomorphic if there exists a collection of isomorphisms of vector spaces Pq: i.e., Aq that intertwine dq and 0 Dq

= dq 0 Pq.

Extrinsic and intrinsic tangential Cauchy—Riemann complexes

135

The following commutative diagram describes part (b) of the definition. DQ

Ag

Aq+i

lPq Ag

Aq+i

Ag+2

As an example, let M and N be smooth manifolds and suppose F: M er+I(M)} and N is a diffeomorphism. The complexes {dM: e"(M) Er+I(N)} are isomorphic and the isomorphism is given by {dN: F*: THEOREM 1

Suppose M is a CR submanifold of

The extrinsic and intrinsic tangential

Cauchy—Riemann complexes are isomorphic. PROOF

Fix p with 0 p n. For q 0, let Ag =

— via the extrinsic definition

Ag =

— via the intrinsic definition.

(M) is the space of smooth Sections of the extrinsically defined bundle which by definition is the orthogonal complement of jp,q ) I M. on the other hand, Ag is the space of smooth sections of the intrinsically defined bundle (M) which by definition is (M)}. We let

Dg: Ag

Ag+i be the extrinsically defined 0M

dq: Ag —4 Aq+i be the intrinsically defined 0M.

= The operator Dg is the tangential part of 0 (i.e., tM o 0), and is the projection of dM where dM is the exterior derivative on M and onto AP{H*'°(M) (M)}. be the inclusion map. We will show that j is the desired isoLet j : M —' morphism between the complexes {Dg: Ag Aq+i } and {dq: Aq ' Ag+i }. The following two statements must be shown:

(i)

(ii)

The map

takes Ag OfltO Ag isomorphically.

j*oDg=dqoj.

To prove (i) it suffices to show the following.

The Tangential Cauchy-Rlemann Complex

136

LEMMA 1

For each point p0 the intrinsic

E

(M) isomorphically onto

M, j maps the extrinsic

For notational simplicity, we will prove this lemma for the case where generic. From Lemma 1 in Section 7.2, we can find an affine complex linear change of coordinates A i—s so that the given point p0 E M is the origin and PROOF

M

is

:

M = {(x+iy,w)

C' x (fl_d;y = h(x,w)}

where h: Rd x Cfl_d Rd is smooth with h(O) =0 and Dh(O) = 0. As mentioned in the remark after the proof of this lemma, A preserves the holomorphic tangent space of M, the totally real tangent space of M and the metric for the (M) real tangent space of M. Therefore, the definition of the intrinsic is invariant under this change of coordinates. In addition, the pull back of A is also commutes with 0. Therefore, the definition of the extrinsic invariant under this change of coordinates. l'he following arguments can be easily modified for the nongeneric case by using the remarks that follow the proof of Lemma 1 in Section 7.2. A local defining system for M is given by {pi,... ,pd} where p3(z,w) = Im —h,(Re z, w). Since Dh(0) = 0, we have .9p3(O) = By definition, the extrinsic is the orthogonal complement in

of the ideal generated by bp1,... OPd. Therefore, a basis for the extrinsic at

the origin is given by

+ IJI = p, IKI = q, I, J, K increasing}.

{dz' A dv? A A basis for L,3 = = is given by

is given by {dwi,.

.

.

,di)Yn_d}. Since

onal complement of L e L in

(M), a basis for

and a basis for is the orthog(M) is given by

{dxi,... ,dxd}. Therefore, a basis for the intrinsic A

A

+

= p, IKI = q, I, J, K increasing}.

Since Dh(0) = 0, the following relations hold at the origin:

1
i*(dx)=dx In

particular, j*(dzk) = dxk for 1

j(dz' A dv? A

k

d and so

= dx' A dv? A

Extrinsic and intrinsic tangential Cauchy-Rle,nann complexes

Hence, j

(at 0) to a basis for the

a basis for the extrinsic

takes

137

(M) (at 0). This completes the proof of the lemma and therefore intrinsic statement (1) follows. I To prove statement (ii), first note from the definitions

Dq =

o

dM

of, statement (ii) will follow from the following lemma.

LEMMA 2

For any integers p, q 71p,q+1

0

T* (C'2)

as maps from

(C'2) } IM

of T (M).

to the intrinsic

For each point p0 E M, the maps in this lemma take the space

PROOF

to the intrinsically defined As with the proof of Lemma 1, we assume M is generic, the point p0 is the origin, and

M = {(x + iy, w) E Cd where h: Rd x Cnd

is

j(dwk)

d?i)k

= dwk, j*dt2)k =

dxk for First, suppose 1

k

= h(x, w)}

x

smooth with h(0) =0, Dh(0) = for 1 k ( n — d and

0.

At the origin,

=

=

d.

is an element of

We have

min(d,p) min(d,q+1) r=O

s=O

I1t=r

a form (at 0) of bidegree (p — r, q + 1 — s) that only involves dWl,...,dWn_d,dlDI,...,dti)n_d. At the origin, tM annihilates dZI,...,dZd.

where each

is

Therefore min(d,p)

>

r0

fl=r

Since fdzk = dxk, we have min(d,p)

r0

(1)

II=r

The Tangential Cauchy—Riemann Complex

138

On the other hand mm(d,p) min(d,q+1)

r=0

III=r

8=0

I

Each

dxk belongs to the intrinsic

J 1=8

and

trinsic

A

A

belongs to the inbelongs to the intrinsic

Due to the presence of

the only contributing term to the sum on the right occurs when s = 0. Therefore min(d,p)

>

r=0

III=r

By comparing this with (1), we see that

= on the space

'T

) I M.

(2)

By examining the above argument, we see that

(this is trivial in the case of

both of these maps vanish on

1) due to the presence of We conclude that (2) also holds on the space and so the proof of the lemma is complete. This completes the proof of statement (ii) and hence the proof of Theorem 1 is also complete. I

j*

0

In view of this theorem and in order to keep the notation to a minimum, we shall not distinguish between the extrinsic and intrinsic OM-complexes. It will be clear from the context which point of view will be used. As the final item in this chapter, we make a remark about metrics. Here, a metric is used to choose a complement, X(M), to A different choice of metric leads to a different X(M) and hence a different ÔM-complex. However, given any two metrics, the associated complementary bundles are isomorphic and therefore the two resulting aM complexes are isomorphic. A metric can be avoided by using quotient spaces. For example in the imbedded case, we may let

= the abstract setting, we let APQ(M) be the space of forms on M of degree p + q that annihilate any (p + q)-vector on M that has more than q-factors In

contained in L. Then we define "

Both of these definitions of

"—

(M) avoid the use of a metric.

Extrinsic and intrinsic tangential Cauchy—Riemann complexes

139

In the imbedded case, the Cauchy—Riemann operator maps the space of smooth sections of jp,q to jp,q+ 1• The tangential Cauchy—Riemann operator can then be defined as the induced map of the Cauchy—Riemann operator on the quotient spaces. Similarly, in the abstract case, the exterior derivative maps the space of smooth sections of to (M). We leave the verification of this to the reader. In this case, the tangential Cauchy—Riemann operator can be defined as the induced map of the exterior derivative on quotient spaces. Both of these complexes are isomorphic to the tangential Cauchy—Riemann complexes defined earlier in this chapter once a metric has been chosen.

We have chosen not to emphasize this point of view because computations usually require a choice of a metric. The metric point of view will be especially useful in Part IV of this book.

9 CR

Functions and Maps

In this chapter, we present the definitions and basic properties of CR functions and CR maps. CR functions are analogous to holomorphic functions on a complex manifold. However, there are important differences. For example, CR functions are not always smooth. There are relationships between CR functions on an imbedded CR manifold and holomorphic functions on the ambient For example, the restriction of a holomorphic function to a CR submanifold is a CR function. However, CR functions do not always extend to holomorphic functions. In this chapter, we show that real analytic CR functions on a real analytic CR submanifold locally extend to holomorphic functions. A C°° version of this is also given. The chapter ends with a discussion of CR maps between CR manifolds.

9.1

CR functions

Suppose (M, L) is a CR struaure. A function f: M distribution) is called a CR function if 0Mf = 0 on M.

DEFINITION I

C (or

For most of this chapter, we shall be dealing with CR functions that are of class C'. The above definition applies to any CR manifold — either abstract or imbedded in We now present other characterizations of a CR function. LEMMA I (a) Suppose (M, L) is a CR structure. only jf Lf =OonMforallLEL.

A C' function f: M —i C is CR if and

pd(Z) = 0} is a generic, CR subman:fold of

CRifandonlyifOfAOplA...AOpd=OonMwheref: extension of f.

140

C is

—iCisanyC'

CR functions

PROOF

141

For the proof of (a), recall that

projection of T (M) onto T only if (dM1, L) =

0

for all L

= 7r°'dMf. Since 7r0'1 is the we have (dM1) = 0 if and

(M) = L , L. Part (a) now follows from the equation

(dMf,7) =12{f} which is the definition of the exterior derivative of a function.

--

(b) follows from the extrinsic definition of OMf as the piece of Of IM which is orthogonal to the ideal generated by {Op; p: CN R is smooth with p=OonM}. I Part

If M is a CR submanifold of

then any holomorphic function on a neighborhood of M in restricts to a CR function on M by part (b) of the lemma. However, the converse is not true, that is, CR functions do not always extend as holomorphic functions. This is fortunate, for otherwise the study of CR functions would be much less interesting. The following example illustrates this behavior.

-

Example 1 Let M = {(z,w) C2; Im z = 0). Here, L = H°"(M) is spanned (over e(M)) by the vector field A function f: M C is CR if

(x= Rez). A CR function on M is a function that is holomorphic in w with x held fixed. Since there is no condition on the behavior of a CR function in the x-variable, an arbitrary function of x is automatically CR. Therefore any nonanalytic function of x is an example of a CR function that does not extend to a holomorphic function on a neighborhood of M in C2. [I In this example, a real analytic CR function on M = {y = 0} is always the restriction of a holomorphic function defined near M. A real analytic CR function on M can be represented (near the origin) by a power series in x and w (no zi). The holomorphic extension is obtained by replacing x by z in this power series. This idea will be exploited in the existence part of the proof of the next theorem, which is due to Tomassini [Tom].

THEOREM I

Suppose M is a real analytic, generic CR subman(fold of

at least n. Suppose f: M

with real dimension

C is a real analytic CR function on M. Then there is a neighborhood U of M in and a unique holomorphic function

F: U—CwithFlM=f.

CR Functions and Maps

142

The neighborhood U in this theorem depends on the CR function f. Additional geometric conditions on M can be added to ensure that the neighborhood U is independent of f. With these added conditions, the real analyticity of f is unnecessary. This and other CR extension topics are discussed in Part Ill. The uniqueness part of the theorem requires that M be generic but it does not require the real analyticity of M. We present it as a lemma. LEMMA 2

Suppose M is a smooth, generic CR subman(fold of C'2 of real dimension 2n — d, o d n. If f is holomorphic in a connected neighborhood of M in C'3 and

tf f vanishes on M, then f vanishes identically. PROOF By the identity theorem for holomorphic functions, it suffices to show

all the derivatives of I vanish at a fixed point po E M. Near p,, E M, there is a local basis for H1'°(M) consisting of smooth vector fields L1,. . (for example, use Theorem 3 in Section 7.2). The collection of vector fields {L1,... ,Ln_d} forms a local basis for H°"(M). Let X1,... ,Xd be a local basis for the totally real tangent bundle, X(M). By ambiently extending the .

coefficients, we may assume these vector fields are defined in a neighborhood ofprj in C'3. The vector fields N1 = JX1,...,Nd = JXd (restricted to M)are transverse to M. Since M is generic, a basis for Tc (C'2) near p0 is given by

{L1,...

,

X1,... ,

,

N1,...

,

The vector fields L1,. . . , L1,. . , X1,.. . , Xd will be called tangential (since their restrictions to M belong to TC (M)). The vector fields N1,. . , Nd will be called transverse. To prove D" f = 0 near p0 on M for all differential operators we use a double induction argument on both the order of the differential operator and the number, rn, of transverse vector fields in If m = 0, then D" involves only tangential vector fields and so D'2f = 0 on M because / = 0 on M. .

.

Now we assume by induction that for rn 0, D" / =

0

on M for all

differential operators that involve only rn-transverse vector fields. We will show

on M where

are

any indices from the set {1,... , d}. From the Cauchy—

Riemann equations on C'3, we have (X + iJX)(f) = 0 for X E T(C'2). nearpo in C'3. Thus Therefore, Njm+L{f} = JXjm+i{f} = N21..

= iN,,..

The right side involves a differential operator with only rn-transverse vector fields and therefore it vanishes on M as desired.

CR functions

To

143

complete the double induction, we assume the following: for integers,

N 0, m 1

D°f=0 on M for N and where involves only rn-transverse vector fields. We also assume f = 0 on M for operators of any order that involve at most (rn — 1)transverse vector fields. We must show that = 0 on M for =N+1 and where involves rn-transverse vector fields. We have two cases to consider.

Case(i). Da=ToDQ'. Here, T is a tangential vector field and is a differential operator of order N that involves only rn-transverse vector fields. In this case, =0 on M by the induction hypothesis and so f} = 0 on M, as desired. = N3 oDa'.

Case (ii) Here,

N3 = JX3 is transverse and

N

is a differential operator of order

that involves only (rn — 1)-transverse vector fields. In this case

= = D°'{N3f} + by ] denotes the commutator. The first term equals , the Cauchy—Riemann equations. This term vanishes on M by the induction hypothesis because D°'XJ involves only (m — 1)-transverse vector fields. The second term is a differential operator of order N and so it vanishes on M, again by the induction hypothesis.

where [

From double induction, it follows that D°f = 0 near p,, on M for all differential operators and so f vanishes identically. The proof of the lemma is now complete.

U

For the existence part of Theorem 1, we give two proofs. The first is per-

haps simpler but the second can be easily modified to handle a Theorem 1. Both proofs illustrate important ideas.

version of

The first proof treats both ( and ( C'1 as independent coordinates. We will show that the given_real analytic CR function f extends to a holomorphic function on C2V1 (of and By using the tangential Cauchy—Riemann equations, we will show that the holomorphic extension of f is independent of the coordinate and so it restricts to a holomorphic function on C'1 which is the desired extension of f. Now we present the details. It suffices to holomorphically extend the given CR function to a neighborhood of a fixed point p0 E M. The global extension FIRST PROOF OF EXISTENCE

144

CR Functions and Maps

can then be obtained by piecing together the local extensions. The uniqueness part of Theorem 1 ensures that the local extensions agree on overlaps. From Lemma 1 in Section 7.2, we may assume the point P0 is the origin and

M = {(z

x + iy, w)

E Cd

x

y

= h(x, w)}

where h: Rd x Rd is real analytic in a neighborhood of the origin and Dh(O) = 0. From Theorem 3 in Section 7.2, a local basis for H°" (M) is given by d

d

the (€, k)th entry of the matrix (I + i(Oh/Ox))'. Since h is a real analytic function near the origin, h can be expressed in a

where

is

power series in the variables z, E Cd and w, E Cn_d. By replacing independent variable ( E Cd and by the independent variable 17 E the power series for h, we obtain a holomorphic function h: Cd x Cd x

= h(x,w). Define

with

Mc = Also define

Cd x Cd x Cd

by the m x

x Cn_d

x

Cd

=

—

x Cd x Cn_d x

by

= Mc is a complex submanifold of with complex dimension 2n — d. Moreis imbedded as a totally real submanifold of Mc. If f: M C is a real analytic function on M, then the above procedure of replacing by ( and ii) by in the power series expansion of / produces a holomorphic function f: Mc —p C with / o = f. Similarly, the real analytic coefficients of L, can be holomorphically extended to vector fields over,

L1,...

T"°(Mc) with

4=

I j n—d.

+

=

Since f is holomorphic, we have

on

then

(1,!)

= on

M.

(1)

If / is CR,

145

CR functions

is a 2n — d real dimensional generic (totally real) submanifold of

Since

we have (2)

on Mc by Lemma 2. Each vector O/O(j is transverse to Mc because Dh(O) 0. Define F: Cd x Cd x Cn_d x C'2_d C that is independent C to be the extension off: Mc So

=

1 j

near the origin,

0

d.

(3)

We also claim

=

0

near the origin,

1

j n—

d.

(4)

To see this, note

a (oP\_ a (a?

—

In view of (1) and using OF/aC, = 0, we obtain

OF

-

-

=0

Since 0/0(, is transverse to

on

Mc,

on

Mc

(by (2)).

the previous two sets of equations imply that

= 0, as claimed. is obtained Finally, the holomorphic extension of f on C'2 = Cd x From (3) and (4), F is independent of ( and and by setting F = F o Therefore, F so the power series of F = F o is independent of and is holomorphic on a neighborhood of the origin in C'2. Moreover FIM

=f

because

FIM = I

=f.

This second proof is based on ideas in a paper by Baouendi, Jacobowitz, and Treves [BiT]. We again start with M presented near the origin as SECOND PROOF OF EXISTENCE

M = {(x + iy, w) E C'1 x Cn_d;

h(x, w)},

CR Functions and Maps

146

where h is real analytic and h(O) = 0, Dh(O) 0. If M is flat (i.e., h 0) then a real analytic CR function near the origin is a convergent power series in x and w (no ff.). The desired holomorphic extension is obtained by replacing x with z = x + iy in its power series. We want to mimic this procedure as much as possible for the general case. The problem is that in general, a real analytic CR function will depend on ti). However, its dependence on ii) is closely linked be with the dependence of the power series of h on ti). Instead of letting

an independent complex coordinate as in the first proof, we shall change the complex structure for C'1 so that h becomes holomorphic. Now we present the details. In the power series of h, we replace x by z. So h is defined on Cd x C'1_d and h(z, w) is holomorphic in z near the origin. Define H:Cd x Cn_d x by

H(z,w) =

(z

+ ih(z,w),w).

be the component functions for H. We use H = (H1,..., Let H1,. . as a coordinate chart to define a new complex structure for C" Cd x C'1. A function g is holomorphic with respect to the new complex structure if there exists a holomorphic function G in the usual sense with g = C o H. This complex structure agrees with the usual complex structure in the z-variables since H is holomorphic in z E Cd in the usual sense. The T°' 1-vector fields for this new complex structure are those vector fields that annihilate the coordinate functions Hi,. . , . ,

.

LEMMA 3 A local basis for the bundle 70,I (C" ) for the new complex structure is given by d

where PROOF

1 <j

is the (i,k)th entry in the d x d matrix [I + i(Oh/Oz)]'.

This lemma follows by showing that for 1 £

n, A,{Ht} = 0 for

= 0 for 1 ( j d. Also note that these

< n — d, and

vector fields are linearly independent near the origin because Dh(0) = 0. Let H0 = HI{Im z=0} The map H0: Rd x Rd x Cn_d be for M. Let 7r: Cd x

—+

I

M is a paraineterization

the projection map given by ir(x + iy, w) = (x, w). Clearly, lrIM is the inverse of H0. From Theorem 3 in Section 7.2, a local basis for H°" (M) is given by d

—

L3

=

—2i

-

#Lk

.9w3

Ozi

+

.9

.9w3

1 j

n— d

CR functIons

147

where Pek is the (4 k)th entry of the matrix [I + i(Oh/Ox)]'. A C' function

f: M —' Cis CRon Mif and only if L,f = 0,1 <j < n—d. This is equivalent to ir.L,{f o H0} 0 on Rd x because H0 o ir is the identity map on M. The vector field L, can be computed by using (O/0y3) = We have 0, = 0/Ox, and =

Suppose f: M Cn_d

—' C.

C is a real analytic CR function. Let fo

The function fo is a real analytic function of x E

Let F0: Cd x Cn_d

Rd

fo

x

and w

be the extension of fo obtained by replacing x by z x + iy in the power series expansion of fo about the origin. Since F0 and h are holomorphic in z Cd, we have Oh/Ozk = Oh/Oxk,OFO/Ozk = OF0/Oxk. we obtain Comparing the expressions for A, and C

AjFoI{Im z=O} =

= 0 (since f is CR). Since A,F0 is holomorphic in z Cd and vanishes on {Im z 0}, A,F0 must vanish identically. Since both A,F0 = 0 and OFO/(92k 0, the function F0 is holomorphic with respect to the new complex structure. There is a function F: C that is holomorphic in the usual sense defined in a neighborhood of the origin with F0 = F o H. The restriction of F to M is I because 1 o H0 = F o H0 on {Imz = 0}. This completes the second proof of the existence part of Theorem 1. We have shown by example that CR functions of class C°° are not necessarily the restrictions of holomorphic functions. However, smooth CR functions are the restrictions of functions that satisfy the ambient Cauchy—Riemann equations on M. This is presented in the next theorem. THEOREM 2

Suppose M is a C°°, generic CR submanifold of C.' with real dimension 2n — d,

1
F defined on such that OF vanishes on M to infinite order and FIM = I. F is unique modulo the space of functions that vanish to infinite order on M.

If M or f is only of class Ck, k 2, then an easy modification of the proof extension, F, such that OF vanishes on M to order k — 1. The key ingredient in the proof of Theorem 1 is the fact that a real analytic function 0: Rd C is locally the restriction of a holomorphic function C' —, C. In the proof of Theorem 2, we must replace this idea with the produces a

following.

148

CR

Functions and Maps

LEMMA 4 lRd (a) Suppose C, d 1, is a function. Then there exists a C°° function '1': Cd C such that O'I vanishes to infinite order on {lm z = O} and 'I = on {lm z = O}. (b) 1 in part (a) is unique modulo the space of smooth functions that vanish to infinite order on {Im z = O}.

Let z = x+iy. The requirement that 'I = on {y= O} determines all the x-derivatives of on {y = O}. The requirement that O'I vanish to infinite order on {y = O} is equivalent to the requirement PROOF

(5)

for all indices Ck. This equation inductively determines all the y-derivatives of

on {y =

Part (a) now follows from the Whitney extension theorem (see Theorem 2 in Section 5.3). For part (b), if '1 vanishes on {y = O}, then all x-derivatives of 1 also vanish on {y = O}. If vanishes to infinite order on {y = O} then from (5) and induction, it follows that all derivatives (in x and y) of vanish on {y = O}. I O}.

The proof of the uniqueness part of Theorem 2 is similar to the proof of Lemma 2. The arguments there show that if F = 0 on M and if OF vanishes to infinite order on M, then all derivatives of F must also vanish on M. PROOF OF THEOREM 2

The proof of the existence part of Theorem 2 is analogous to the second proof of the existence part of Theorem 1. We sketch the ideas. The graphing function h for M (and hence H) can be extended to Cd x as smooth functions so that 02h and vanish to infinite order on {Im z = 0} by Lemma 4.

As before, we use H = (H1,...,

to define a new complex structure for Lemma 3, which exhibits a local basis for the T°" bundle of this new complex structure, is still valid. For a smooth CR function / on Extend fo to F0: Cdx Cn—d that M,let fo foHl{Jm vanishes to infinite order on {Im z =O}. The computation that A3F0 vanishes on {Im z O} is still valid. Since and vanish to infinite order on {Im z = 0}, also vanishes to infinite order on {Im z = 0}. By part (b) of Lemma 4, A,F0 vanishes to infinite order on {Im z = 0}. Hence, both A,F0 and vanish to infinite order on {Im z = 0}. Since H: Cd x Cd x Cn_d is a diffeomorphism near the origin, we can define

F=F0oH*

f

As before, FIM = f because F o HI{Im z=0} = fo = 0 It remains to show that vanishes to infinite order on M. This is equiv-

CR maps

149

alent to showing IF vanishes to infinite order on M for all I '(Ca) for the usual complex structure for

to

belonging

We already know that

IF0 vanishes to infinite order on {Imz = O} for all I in the bundle for the new complex structure for Since H i—' C" is the coordinate chart for the new complex structure for C", the push forward map under H' sends the T°" bundle for the usual complex structure for C" to the bundle for the new complex structure for C". Thus, vanishes to infinite order on {Imz = O} for all L in the bundle for the usual complex structure for C". Since F = F0 o we conclude that LF van-

H',

ishes to infinite order on M for all I in the T0" bundle for the usual complex structure for C", as desired. The proof of Theorem 2 is now complete. I

9.2

CR maps

Suppose M and N are CR manifolds and f: M —' N is a C' map. If the target space N is a CR submanifold of C"', then f has component functions (I,,... fm). In this case, it is reasonable to call I a CR map if each of the ,

component functions is a CR function. This definition needs to be modified in the case where N is not imbedded in Ctm. To motivate the abstract definition, let us more closely examine the case where both M and N are CR submanifolds

of C" and Cm. Using Theorem 2 in the previous section, we can extend M N to a function F = (F,,... , Fm) defined on C" f = (ft, ... , so that = 0 on M, 1 j $ m (here, all we need is that OF? vanish to first order on M). So for p E M, maps into In addition, F.(p) maps into to because F maps M to N. Therefore, for p E M, maps and

into

Hz,) (N) and Hg" (M) into H°F.(P) (N). For an abstract CR structure (M, L),

the subbundle L takes the place of H"°(M). This motivates the following definition. DEFINITION 1

f: M

Suppose (M, LM) and (N, LN) are CR structures. A C' map

N is called a CR map if f,, {LM } c LN.

The extension of

to T' (M) satisfies f.(I) =

(this is true for any C1 map). Therefore, Definition requirement that

C

1

In particular, f.{LM

for any L E Tc (M) is equivalent to the

c LN eLN.

LEMMA I

Suppose (M,L) isa CR structure. A C' map f = (f,,. .. a CR map if and only

each

is a CR function.

M '—p Ctm is

CR Functions and Maps

150

Here, the target space is the CR structure (Ctm is CR if and only if belongs to for each L

The map f

PROOF

L. For 1

j 5 m.

let zj be the jth coordinate function for Cm. We have of =

o f}

= L{f3}. if and only if L{f3} = 0 for 1 j ni. The proof of the lemma now follows from part (a) of Lemma 1 in belongs to

It follows that Section 9.1.

I

Define the subbundle

H(M) = {L+L; L E This is a subbundle of the real tangent bundle to M. From Lemma 3 in Section 3.2, there is a complex structure map J: H(M) —i H(M) so that the extension of J to Hc (M) = IL €i L has eigenspaces L and L corresponding to the eigenvalues +i and —i. The following theorem gives an alternative charon H(M) and the J acterization of a CR map in tenns of the action of map. THEOREM I

H( M) H (Al) Suppose (M, and (N, LN) are CR structures. Let and JN: H(N) —p H(N) be the associated complex structure maps. A C' map

f: Al

N isa CR map land only if for each p E M,

C

and on

One of the characterizations of a holomorphic mapping between two complex

manifolds is that the derivative commutes with the complex structures. The point of Theorem I is that the analogous characterization holds for CR maps between CR manifolds. Theorem 1 is sometimes summarized by saying that The reader should is a complex linear map from to not confuse this meaning of complex linear with the concept of a complex linear map between two complex vector spaces such as (M) and (N). The

push forward of any C' map f : M to

N is complex linear as a map from

(see Section 3.1).

PROOF OF THEOREM 1

Let us first assume that I is a CR map as in Defini-

tion 1. For L E LAI

+ L) = f1(L) + f1(L).

151

CR maps

(by Definition 1), clearly is an element of H(N). e In addition, LM and LM are the +i and —i eigenspaces for JM. Therefore

Since

11 (JM(L +

1;))

=

—

=

iL) —

Since (L) is an element of LN, which is the +i eigenspace of JN, the above equation becomes

JN(f.(L+T)). Thus, f. o

on H(M), as desired. = JN o For the converse, note that each element L in LM can be written as

L=X-jJMX where

X = l/2(L + L) E H(M). We have

f.(L) =

—

=

—

if*(JMX)

LN is generated by vectors of the form Y — iJNY for Y E H(N). Since f.(X) belongs to H(N), the above equation shows that belongs to LN, as desired. The proof of the theorem is complete. I Now we turn our attention from the push forward of vectors to the pull back

of forms via a CR map. If f is a holomorphic map between two complex manifolds then f* commutes with a. l'his must be modified for CR maps with the tangential Cauchy—Riemann operator. If (M, L) is a CR structure, then the tangential Cauchy—Riemann complex involves the totally real part of the cotangent bundle, X*(M) as well as L* and L*. However, the definition of a CR map makes no requirement about the behavior of on X(M). So

unlike the case for holomorphic maps, the pull back operator of a CR map, 7, does not preserve bidegree, i.e., is not necessarily contained in (see Lemma 2 below). Therefore, f* does not quite commute with the tangential Cauchy—Riemann operator. For example, let M = {(z = x+iy, w) E M is given by f(x,w) C2; y = O}. Suppose f : M (x,w + x). Clearly, f is a CR map because the component functions of I are holomorphic n w. Note that diii + dx. We also have = (9M(W±X) = diii, and so (9Mf*(w). Instead, the following equation

holds: OMf(iiJ) = theorem.

More generally, we have the following

CR Functions and Maps

152

THEOREM 2

Suppose M and N are CR mantfolds and f: M —b N is a CR map. Then

of' = as maps from

of'

to

We remark that if the metric-free version of the tangential Cauchy—Riemann complex (defined at the end of Chapter 8) is used, then the pull back operator via a CR map preserves the bidegree. In this case, the tangential Cauchy—Riemann

operator commutes with the pull back operator via a CR map. We leave the verification of this to the reader. As mentioned in Chapter 8, computations usually require the choice of a metric. For this reason, we have chosen to show how the tangential Cauchy—Riemann complex defined via a metric behaves with respect to pull backs of CR maps. As an example, let M be a CR submanifold of and suppose j : M C is map. From Theorem 2, we have

= equation relates the intrinsically defined tangential Cauchy—Riemann operator to the ambient 8-operator. The proof of Theorem 1 requires a preliminary result. This

LEMMA 2

Suppose F: M

N is a CR map between CR structures (M, LM) and

(N,LN). Then (a)

F'{T"°(N)} c T"°(M)

(b)

Forp,q0 C

where r = min{q, n — p} with n PROOF

e dimc T"°(M).

For the proof of part (a), let 4> E T"°(N). We must show

(F'4>,r) =0 for all L E T°"(M) = EM where (, ) denotes the pairing between forms and vectors. Since F is CR, F.L is an element of LN = T°"(N) and so 0 = (4>, F,L) = (F'4>, L), as desired. Part (b) follows by writing a typical term in (N) as

where

Note that

E

T"°(N)

and E T'°"(N) and then by using part (a) for generally has nontrivial components of type (1,0) and (0,1).

I

CR maps

153

PROOF OF THEOREM 2

Let 0 be an element of

= F0 —

From Lemma 2, we have

+

I

From the definition of t9M, we have

_JF*0} belongs In view of Lemma 1 in Section 8.2, 9+J'Q—3+l Since j 1, the sum on the right must vanish. Therefore

= Using the fact that dM commutes with F*, we obtain

=

(1)

From the definition of 0N and Lemma 1 in Section 8.2, we have dNO = '9N0 + By Lemma 2,

+

(2)

isa sum of terms of type (p+ 1 +j,q—j) for Inparticular

= 0. Similarly

=

0.

These two equations together with (1) and (2) yield

=

The proof of Theorem 2 is complete.

I

We discuss two corollaries. First, we give the extrinsic version of Theorem 2 for imbedded submanifolds.

COROLLARY I

Suppose M and N are CR submantfolds of and respectively. Suppose f: M N is a CR map. Let F: —i Ctm be an extension off with OF = 0 on M. Then as maps from the extrinsically defined

to the extrinsically defined

CR Functions and Maps

154

and ON refer to the extrinsically defined tangential Cauchy—Riemann

Here,

operators. Elements of are not intrinsic to N. Rather, they are smooth seclions of For this reason, it is necessary to have C an ambient extension (F) of the CR map (f) for the statement of the corollary.

One approach to the proof is to show that Ft maps jp,q IN to M (see the definitions of these spaces in Section 8.2). Then, the corollary will follow from the definition of the extrinsic version of the tangential Cauchy— PROOF

I

Riemann complex. The other approach to the proof involves reducing the statement given in the corollary to Theorem 2. We will give the details of the latter approach and leave the details of the former approach as an exercise. Let :M and iN; N be the inclusion maps. By Section 8.3, and are isomorphisms between the extrinsic and intrinsic tangential Cauchy—Riemann complexes of M and N. Therefore, the statement of the corollary is equivalent to

OtM oF Since bF =

as operators on the extrinsically defined serves bidegree for elements of

0

on M, F pre-

and so this equation is equivalent

to o

o Ft =

0 tM 0

o

0

oFo

From Theorem I in Section 8.3, and ON = °9M = ÔM ) 'oON where the OM and ON on the left are extrinsic and the 8M and ON on the right are = intrinsic. In addition, from Lemma 2 in Section 8.3. Therefore, the above equation is equivalent to o

o (F

OjM) =

o (F ojM) o

o

OM and ON are now the intrinsic tangential Cauchy—Riemann operators. o f (since F = Finally, we use the fact that F o on M) to see that = this equation is equivalent to Here,

f

of OON) is an isomorphism between the extrinsic and intrinsic Eu', the proof of the corollary now follows from Theorem 2. I Since

We say that the CR structures (M, L) and (N, LN) are CR equivalent if there is a CR diffeomorphism between M and N. Using Theorem 2, we will show (in Corollary 2) that if M and N are CR equivalent, then the tangential Riemann complex on M is solvable if and only if the same is true for N. To be precise, we say that the tangential Cauchy—Riemann complex is solvable at bidegree (p, q) if for any form f E with OMf = 0, there is a form U E

with OMU =

f.

CR maps

155

COROLLARY 2

Suppose (M, LM) and (N, LN) are equivalent CR structures. The tangential Cauchy—Riemann complex is solvable at bidegree (p, q) on M if and only if the

is true for N.

Since an open subset of a CR manifold is also a CR manifold, the above corollary applies to CR equivalent open subsets of CR manifolds.

Suppose F: N is a CR diffeomorphism and suppose f is an element of with 0Nf = 0. By Theorem 2, we have PROOF

=

=0 on

M.

If the OM-complex is solvable at bidegree (p, q) on M, then there is a form uE

with OMU =

Applying

o

F'

to this equation and using Theorem 2 with M replaced

by N and F replaced by F', we obtain =

(3)

From Lemma 2, we have

= where

r

Ff

-

min(q, n — p). Applying

o

F'

to this equation and using

Lemma 2 with F-' instead of F, we obtain = — —

p,q

=1 Substituting this equation into the right side of (3) yields on

N.

Hence, the solvability of OM implies the solvability of ON. The converse is established the same way. I

10 The Levi Form

In previous chapters, concepts such as the tangential Cauchy—Riemann complex are introduced first for imbedded CR manifolds and then later for abstract CR

manifolds. In this chapter, we take the opposite approach. First, we give the definition of the Levi form for the case of an abstract CR structure and then proceed to give more concrete representations of the Levi form in the case of an imbedded CR manifold. The Levi form for the case of a real hypersurface in is discussed in some detail. In particular, the relationship between the Levi form and the first fundamental form of a hypersurface is presented.

10.1

Definitions

One of the defining properties of an abstract CR structure (M, L) is that L is involutive (i.e., [L1, L2] E L whenever L1, L2 E L). The subbundle L L C Tc (M) is not necessarily involutive. In fact, the Levi form for M is defined so that it measures the degree to which L L fails to be involutive. For p E M, let

0 C —+

0 C}/(L,

be the natural projection map. DEFINITION I

L

156

The

Levi form at a point p E M is the map

in L that equals

at p.

157

DefinitIons

The vector field L] lies in Tc (M) since Tc (M) is involutive. So the Levi form measures the piece of (1/2i)[L, that lies "outside" of IL,, e The factor 1/2i is introduced to make the Levi form real valued, i.e., = In order to show that the Levi form is well defined, we must show that its definition is independent of the L-vector field extension of the vector

L and Z are two vector fields in L with

= Zr,, then

=

ir,,[t,

Fix p EM and let {L1,. . ,Lm} be a basis for L that is defined near p. For some unique collection of smooth functions at,... , am and b1,... , bm, PROOF

.

we have

L= near

p.

The assumption that = Z,, means that a3 (p) = bj (p) Expanding the Lie bracket, we obtain

for 1 j

m.

[t,L] = =

a,ak[L3, Lk] mod

(LeE).

j,k=I Therefore

= a

j,k=1 Since

a3(p) = b3(p), the proof of the lemma is complete.

I

1fF: M NisaCRmapbetweentheCRstructures (M,LM)and(N,LN), Therefore, F.(p) induces a then F.(p) maps (LM eLM)P to (LN map on the quotient spaces

{Tp(M)®C}/(LM

{TF(p)(N)®C}/(LN

The Levi Form

158

LEMMA 2

Suppose (M, LM) and (N, LN) are CR structures and let £M and £N be their respective Levi forms. If F: M —' N is a CR d(ffeomorphism, then for p E M

Ft'I\l ° ,M_,N 1' — as maps from {LM

0

L'f

L

to {Tp(p)(N) ®

LN)F(p).

PROOF The proof follows from the definitions and the observation that I = say a CR structure (M, L) is Levi flat if the Levi form of M vanishes Im z = O}. at each point in M. For example, let M = {(z, w) C x Since A (global) basis for L = H"°(M) is given by 0/Owi,... = 0, M is Levi flat. Also note that M is foliated by the We

complex manifolds

for xER. The complexified tangent bundle of each following more general result.

is given by L

We have the

THEOREM 1

Suppose (M, L) is a Levi flat CR structure. Then M is locally foliated by complex manifolds whose complexified tangent bundle is given by L e L.

PROOF The idea of the proof is to show that L L and its underlying real bundle are involutive. Then the foliation is obtained by the real Frobenius theorem. The Newlander—Nirenberg theorem will then be used to show that the — submanifolds in the foliation are complex manifolds.

L and L are involutive by the definition of a CR manifold, L ® L is involutive if and only if [L1, L2} is an element of L e L whenever L1 and L2 and belong to L Since Mis Levi flat, [T1,L1], +Z2,L1 +L21 belong to L e E. After expanding [7i + L2, L1 + L2], we see that Since

A

belongs to LeL. A similar computation involving that the vector field

+iL2, L1 +iL2} implies

B =[7.1,L2]+[i.1,L2} also belongs to L E. Adding A and B, we see that [L1, L2] belongs to L E and so L L is involutive. The underlying real bundle for L L is the space

H(M)={L+T; L€L}.

The Levi form for an imbedded CR manifold

159

is involutive, H(M) is an involutive real subbundle of T(M). The Since real Frobenius theorem (Section 4.1) implies that M is foliated by submanifolds, {M'}. such that for each p E M'.The complexified tangent Li,. Since L is involutive, space for M' at p is given by 0C= (M', L) forms an involutive almost complex structure. By the Newlander— Nirenberg theorem, there is a complex manifold structure for M' so that the is L This completes the proof of the theorem. resulting The reader should note that if M is a Levi flat CR submanifold of C1', then the easier imbedded version of the Newlander—Nirenberg theorem (Theorem 2

in Section 4.3) can be used to show that each leaf of the foliation, M', is a complex submanifold of C1'.

10.2

I

The Levi form for an imbedded CR manifold

Computations with the Levi form are facilitated by identifying the quotient space This is accomplished with a subspace of

by choosing a metric for Tc (M) and then by identifying the quotient space ® L1, with the orthogonal complement of IL,, L, (denoted ® in Section 8.2).

For an imbedded CR manifold M, a natural metric exists — namely the restriction of the Euclidean metric on TC (C') to Tc (M). In this case, L = H"°(M), L = H°"(M) and the quotient space

is identified with the complexified totally real part of the tangent bundle. As = and so mentioned earlier, the Levi form is real valued, i.e., which is the totally real part the image of the Levi form is contained in of the real tangent space of M at p. With this identification, the Levi form of a CR submanifold, M, at a point p E M is the map —*

given by

L(

\l

p) —

'-'p

1,0 p

is the orthogonal projection and where L is any H"° (M)-vector field extension of the vector Sometimes, it is convenient to think of the Levi form of an imbedded CR which manifold as a map into the normal space of M at p. denoted where irk:

The Levi Form

160

is the orthogonal complement of in composing £,, with J and then projecting onto

This is accomplished by Let

be the orthogonal projection map. The extrinsic Levi form of Al at p is the map given by *, o J o 4).

DEFINITION I

Since

are J-invariant, we have

and

= where

L is any H"°(M)-vector field extension of 4,.

Now, we develop a formula for the Levi form in terms of the complex hessian of a set of defining functions for M. THEOREM 1

Suppose M = .. = Pd(() = O} isa smooth CR subman(fold E with 1 d n. Let p be a point in M and suppose {Vp1 (p) of Then the extrinsic Levi form is Vpd(p)} is an orihonormal basis for given by

4(W) = for W =

Wk(o/ock)

Vp1(p)

E

PROOF We start with the definition of

= for W E H"°(M). Since {Vp1 (p),. . . , Vpd(p)} is an orthonormal basis for —+ N,(M) is given by the projection

= where ( , ) denotes the pairing between one-forms and vectors. So the £th component of 4) (Wy) is given by

J[W,

The Levi form for an i,nbedded CR manifold

Using

the dual of J, denoted J':

161

we obtain

—'

=

£th component of

1W,

Recall thatd=O+8; J*oe9=iO and J*oO=_ie9. Therefore =

£th component of

—

9)pt(p), [W,

(1)

Now we use the formula for the exterior derivative of a one-form in terms of its action on vectors (see Lemma 3 in Section 1.4). For a one-form and vector fields L1, L2, we have

AL2) =

—

L2

W E H"°(M). We have

=

—

W is of type (1,0) and Opt is a form of the bidegree (0,1), the second term on the right vanishes. The first term on the right also vanishes in view of Lemma 2 in Section 7.1. Therefore, we have L2) = 0. Similarly, we have L1) = 0. Equation (2) becomes Since

W) =

—

—((a — b)p1, 1W, W)).

Comparing this with (1) yields

=

£th component of

=

—

(since d=O+O)

= for

=

wk(O/O(k). The proof of the theorem is now complete.

I

Theorem I is often used in conjunction with Lemma 1 in Section 7.2. In that

lemma, the point p M is the origin and coordinates ç = (z = x + iy, w) E Cd x are chosen so that the defining functions for M are pt(z,w) = — h1(x, w), 1 £ d, with ht(0) = 0 and Dht(0) = 0. Note that Vpt(0) = and so {Vp1 (0),. . , Vpd(0)} is an orthonormal basis for No(M). We identify No(M) with via the map y = (y',... ,yd) i—' E .

The Levi Form

162

with Cn_d by the map

We also identify

n-d t9Wk

k=I

With these identifications, the restriction of the action of the complex hessian of

is the same as the action of the (w, Pt to the directions in of pi on d• We obtain the following corollary. COROLLARY I

is smooth and h(O) = 0, Dh(O) = 0. Then the extrinsic Levi form a: 0 is given by

n—d

lit

= 4i

j,k=I

for W = (w1,.

ÔWJthZ'k

W3Wk

E

This corollary can also be established by expanding ILk, L31 where L1,..., Lnd is the local basis for H"°(M) given in Theorem 3 in Section 7.2. Let us specialize to the case of a quadric submanifold Al = {y = q(w, ti)}, where q: Cn_d x Cd is a quadratic form (Definition 1 in Section 7.3). From Corollary 1, the Levi form of M at the origin is given by

k

In other words, the Levi form at the origin of a quadric submamfold is the associated quadratic form q (restricted to {(w, zn); w E Theorem 2 in Section 7.3, which describes a normal form for codimension two quadrics in C4, can now be interpreted in terms of the Levi form. In this

case, No(M) is a copy of R2 and Hd'°(M) is a copy of C2. In part (a) of that theorem, M = {y' = q1(w, ii)), Y2 = 0} and the image of the Levi form of M at 0 is contained in a one-dimensional line (the Yl axis) in R2. In all three cases in part (b), the image of the Levi form is a two-dimensional cone in R2. In case (i), M = {y' = 1W112,Y2 = 1w212} and the image of the Levi form is the closed

quadrant {yi O'Y2 0}. In case (ii), M = {y' =

!w112,y2

and the image of the Levi form is the open half space > 0} together with the origin. In case (iii), M = = Re(w1z12),y2 = and the image of the Levi form is all of R2 No(M). In all of these cases, the image of the Levi form is a convex cone in N0(M). In general, the image of the Levi

The Levi form of a real hypersurface

163

However, the image of the Levi form is not always form is a cone in convex, as we shall see in example (v) m Section 14.3.

103 The Levi form of a real hypersurface One of the best studied classes of CR manifolds is the class of real hypersurfaces

So we shall devote a section to the study of the Levi form for a real —+ JR is hypersurface in Let M = {z E = 0}, where p: smooth. If p is a point on M with IVp(p)I = 1, then from Theorem I in Section 10.2, the extrinsic Levi form is given by in

n

In this case, is isomorphic to wk(O/thk) a real line via the map t '—i tVp(p), t E JR. For this reason, Vp(p) is often for W =

dropped and the Levi form is then identified with the restriction of the complex hessian of p to The above formula requires JVp(p)I = which can always be arranged by multiplying p by a suitable scalar. However, it is important to note that if is 0 on M, then the map another defining equation for M with 1

for j,k=1

a nonzero multiple of the Levi form at p. To see this, first note that ,5 = ap —+ R, which is nonzero near M (see Lemma 3 for some smooth function a: in Section 2.2). Therefore is

n

-

O(J8(I,

n

= a(p)

+2Re

W3tiJk

{(E Op(P) )

+p(p)

(

(k

The third term on the right vanishes because p(p) = 0 for p E M. The second w,(a/th2) E = 0 for W = term also vanishes because H"°(M) (by Lemma 2 in Section 7.1). Therefore, the complex hessian of

The Levi Form

164

differs from the Levi applied to the vector W = E In particular, information about the Levi form form of M at p by the factor such as the number of nonzero eigenvalues can be determined by examining the complex hessian of any defining function for M. A special case worth examining occurs when the Levi form is definite.

A real hypersurface M is called strictly pseudoconvex at a point p EM the Levi form at p is either positive or negative definite, i.e., if there exists a defining function p for M so that DEFINITION 1

n

F-so-p (,

j,k=1

for all W =

(p)w,wk -

>0

w3(O/O(3) E

The above inequality is an open condition. Furthermore, it is invariant under a local biholomorphic change of coordinates. This follows by explicitly computing the complex hessian of poF where F is a biholomorphism or by using Lemma 2 in Section 10.1.

A real hypersurface M is called strictly pseudoconvex if M is strictly pseudoconvex at each point p E M. THEOREM 1

Suppose M C is a smooth real hypersurface that is strictly pseudoconvex at a point p E M. Then there is a biholomorphic map F defined on a neighborhood U of p in C'2 so that F{MflU} is a strictly convex hypersurface in F{U} C C'2.

PROOF The idea of the proof is to holomorphically change variables so that the real hessian of the defining function in the new variables is positive definite. First, wechoosecoordinates (z,w) CxC'2' asinLemma 1 inSection7.2 so that p is the origin and

M = {(z = x+iy,w) E Cx

= h(x,w)}

—i R is smooth and h(0) =0, Dh(0) =0. By Theorem 2 where h: R x in Section 7.2 (with k = 2), we may assume there are no second-order pure

terms in the expansion of h about the origin, i.e., 02h(0)

—

OWjOWk — ÔX,OWk

Let

—0 —

p(z,w) = y— h(x,w). We have

+ 0(3)

p(z, w) = y + j,k=1

Ic

The Levi form of a real hypersurface

165

where 0(3) denotes terms that vanish to third order in x and to (i.e., 10(3)1 C(1x13 + 1w13). We may assume the quadratic expression in w and tD is positive definite (the negative definite case is similar). Now, we modify p and make a holomorphic change of coordinates so that the quadratic piece of the defining function in the new coordinates is positive definite in z, 2 as well as w, ü). Let

,5=p+2p2. Note that

is

also a defining function for M. We have

w) = y + 2y2 +

WiZDk + 0(3)

()

=y— Re(z2)+1z12+

(1)

Define the following change of variables, (2, ti,) = F(z, w)

2=z—iz2,

zEC wE

F is a local biholomorphism which preserves the set {(O, w); w E

2=+

}.

If

then

Re(z2)

= 1z12 + 0(3).

Let M =F{M} and let tion forM is given by

(2)

(2, ti,)) = 15(z, w). A defining equa= 0. Using (1) and (2), we obtain n—i

5 is positive definite in 2, zli. Therefore, M = I {(2,t1);,3(2,tl,) = 0} is strictly convex in a neighborhood of the origin.

If the complex hessian of the defining equation of M is only positive semidefmite on for each p EM, then the hypersurface is called pseudoconvex. The analogue of Theorem 1 does not hold for pseudoconvex hypersurfaces. There is an example (see [KN]) of a real hypersurface that is strictly pseudoconvex everywhere except at one point p0 which is not biholomorphic near p,, to any (weakly) convex hypersurface in

166

The Levi Form

We conclude this chapter with the comparison of the Levi form of a real hypersurface in with its second fundamental form. Our presentation is similar to that in [Tail. First, we review the definition of the second fundamental

form. Suppose M is a real hypersurface in RN that locally separates RN in two open sets D and RN — D. Let N be the outward pointing unit normal vector field to D on M. We assume M is locally oriented according to N, which means that a collection of vectors Xi,..., XN_I in is considered positively oriented if X,,... , } has the same orientation as RN. Suppose W = w3 is a vector field on RN and let be an element of TP(RN). Define the vector by E

In other words, Vv9W is the derivative of W in the direction of and W are vector fields, then [V, = — Vw9V. DEFINITION 2 The second fundamental form is the map lIp: IR defined by

=

.

W,,

V

x

for

where (.) is the Euclidean inner product on RN.

In the next lemma, we derive a formula for the following defining function for M

p(x)

If M is

then p is

I —dist(x, M)

if x if x

= 1 dist(x, M)

in terms of the real hessian of

D RN

—

D.

near M in RN and Vp = N on M.

LEMMA 1 Let p M, and suppose V, and

W are

=

(a)

(b)

respectively. Then

and

If

=

and W,, =

wk(a/Oxk), then N

crp(p) OxOx j,k=1

Ic

V,Wk.

The Levi form of a real hypersurface

167

For part a), note that N• W = 0, since W E T(M) and N is the unit normal. From the product rule, we have PROOF

V{N.W} =(VvN)W+N.(VvW)

0

and part (a) follows. For (b), write N = Vp = >2(Op/Oxk)(ô/ôxk). Then

= Taking the inner product of this vector with

yields the formula in part (b).

Note that part (b) shows that 1I,,(.,.) is a symmetric bilinear form. To compare the Levi form with the second fundamental form, the first problem to overcome is that ii,, is defined on the real tangent space whereas is defined on which is a subspace of the complexified tangent space of Al. If then W,, X,, — iJX,, where X,, = is an element of + W,,) E C T,,(M). If X is a H(M)-vector field extension of X,,, then W X — iJX is a H"°(M)-vector field extension of We have

=

+iJX,X — = So for the purposes of comparing and let us identify — iJX,,. Then £, is identified with the map = for X,, E The projection —. is given by

with W,, = (V

.

Therefore

= —J[X,

N,.

The Lie bracket [X,JX] can be expressed as Vx(JX) — have J(Vx Y) = Vx JY by explicit computation. Therefore

= —J{Vx9(JX)

—

=

+

We also

.

(from Lemma 1).

Note that if

is a vector in the J-invariant subspace C T,(M) and therefore 119(JX9,

also belongs to We have established the following theorem.

then JX, is well defined.

The Levi Form

168

THEOREM 2

where

X,, E

is identified with form.

=

—

iJX,

E

(M) in the definition of the Levi

If M is convex, then the second fimdamental form is positive (or negaand tive) semidefinite. If = 0 then by Theorem 2, both and are null vectors must vanish. In this case, both (i.e., in the 0-eigenspace) for the second fundamental form. Therefore

=

Y,) = 0 for all

We have

0= (3)

for all Y,,

Now for any X,,

is

an element of

because

= 0. Likewise, we have = 0. This fact Therefore (3) implies is has the following geometric interpretation for a convex hypersurface: if a null vector for the Levi form then the derivatives of the unit normal vector field in the directions of X, and JX, both vanish.

11 The_Imbeddability of CR Manifolds

In Section 11.1, we show that any abstract real analytic CR structure is locally CR equivalent — via a real analytic CR diffeomorphism — to a generic, real analytic CR submanifold of C's. Nirenberg's C°° counterexample presented in Section 11.2 shows that without additional hypothesis, the corresponding theorem for C°° CR structures is false. Additional imbedding results will be discussed in Chapter 12.

11.1

The real analytic imbedding theorem

Recall that the CR codimension of M is the number d = dimc {Tc (M)/LeL}. then the CR codimension of M is the If M is a generic CR submanifold of same as the real codimension of M. To say that a CR structure (M, L) is real analytic means that M is a real analytic manifold and that L is a real analytic subbundle of Tc (M), i.e., L is locally generated by real analytic vector fields. Now we state the imbedding theorem, which first appeared in [AnHil]. THEOREM I

Suppose (M, L) is an abstract real analytic CR structure with CR codimension = d 1. Given any point p0 M, there is a neighborhood U of p0 in M so that (M fl U, L) is CR equivalent via a real analytic CR map to a generic real analytic CR subman(fold of complex Euclidean space with codimension d.

One of the defining properties of a CR structure (M, L) is that L is an involutive subbundle of TC (M). One might be tempted to think that this theorem should follow from the real Frobenius theorem (which does not require the real analyticity of L). However, the real Frobenius theorem requires the underlying real subbundle of L (i.e.,H(M) = {L + L; L L}) to be involutive. If H(M) is involutive, then L L is involutive. This is equivalent to saying that M is

169

The Imbed4ability of CR

170

Levi flat which is not assumed here. Instead, we shall use the real analyticity of M and L to complexify L and then use the complex analytic version of the Frobenius theorem. Suppose that m = dimc L = dmc E. Near p0 in M, L is generated PROOF by rn-real analytic vector fields {L1,. .. , Lm}. Since d is the CR codimension of (M,L), we have 2m + d = dinIcTC(M) = dilnR(M). Using a local real analytic coordinate system for M, we may assume that M is an open subset of R2m+d containing the origin and that each L3 is a real analytic vector field in 7'C (R2m+d). Denote the coordinates of We Write by (UI,.. , .

2rn+d

a real analytic, complex-valued function of u

Since

{Li,...,Lm} is linearly independent, the matrix (a3k(O)) I 5 m,
,

m+d

I 5j

= k=I

Here, each A,k

is

5m.

k

a real analytic, complex-valued function of u = (t, x)

UC

]R2m+d

The Lie bracket [L,, Lk] has no (O/&te)-component. On the other hand, [L3, Lk] is a linear combination of {L1,..., because L is involutive. Any nontrivial linear combination of {L1,... , Lm } contains a nontrivial linear comTherefore, we conclude bination of O/t9i1,.. . ,

[L3,Lk]=O Now we complexify each L2. Let ( E Ctm and z E Cm+d be the complexifications of t E Rm and x Rm+d respectively (so Re ( = t and Re z = x). By replacing t and x by ( and z in the power series expansion of (about 0), we obtan functions which are holomorphic in a neighborhood C U of the origin in C2m+d with )t3k(t,X) = Define

=

—+

m+d k=I

lSj Sm.

The real analytic irisbedding theorem

171

z) is holomorphic in ('and z, we have

Since

f9Ajk(t,x)

OA,k(t,x)

—

OA3k(t,X)

and

—

Together

—

with [ti, Lk] = 0, we obtain EL,,Lk](t,x) = 0

for

(t,x) EU

C

From the identity theorem for holomorphic functions, we obtain

—

From Lemma 1 in Section 4.2 (with n = 2m+d), there is a holomorphic map Cm+d cm+d which is defined on a possibly . , Zm+d): Ctm x

Z = (Z1,. smaller

.

neighborhood U of the origin in C2m+d, so

that

on U Z(0,z)=z Here, we are

only

(0,z)EU.

for

using the last (m + d)-componenls of the map Z:

x

Cm+d

C2tn+d given in this lemma. Define the real analytic map Z: U fl ]Wn+d} Cm+d by x {Rtm

= Z(t,x)

Z(t,x)

for

(t,x) ERtm x

We have

=0 on U n {Rm x Rm+d}

= In

addition

Z(0,x)

We

claim that Z

Since

is

our desired imbedding.

m, 1 k ( m + d, Z is

L, Zk = 0 for 1 j

in Section 9.2 and

Lemma 1

derivative at the origin

maximal

= Z(0,x) = x for (0,x) EU.

rank.

Write

in Section

of Z as

Z(t,x) =

9.1.

a CR map by Lemma I

So, it suffices

to

show that the real

R2m+2d has a map from R2m+d to Cm+d U(t,x) + iV(t,x). We must show that the

matrix ÔU

/

(

M=I

8V

\ has real rank 2m + d.

Since

x = Z(0, x) = U(0, x)

Ox

Ox

+ iV(0, x), we

have

The hnbed4ability of CR Manifoids

172

= 0 can be described in the follow-

The imaginary part of the equation ing matrix form:

Ov

IOV\

IOU\

=0.

Here, Re A and Im A are the m x (m + d) matrices whose (j, k)th entry is Re A3k(t, x) and Im Ajk(t, x), respectively. By evaluating this equation at the origin and using (ÔU/ôx)(0) = I and (OV/ôx)(0) = 0, we obtain = —(Im A)t(0).

Therefore, the matrix M has maximal rank if and only if Im A(0) has rank m. It is here that we use the assumption that L fl L = {0} (from the definition of a CR structure). This implies that the set {L1 — L1,.. . ,Lm — Lm} is linearly independent over C. For 1 j m, the term

-

m+d

=

Im A2k

is essentially the jth row of the matrix Im A. It follows that Im A(0) has rank m and so M has maximal rank, as desired. Therefore, there is a neighborhood U of the origin in such that

= Z{U} The real codimension of M' in Cm+d is d because dima M' = dlmR U = 2m + d. The proof of Theorem I is now is a real analytic submanifold of complete.

11.2

I

Nirenberg's nonimbeddable example

In this section, we present Nirenberg's example [NIT] which shows that the imbedding theorem fails for CR structures. Our treatment of this example is taken from [JT]. Nirenberg's example is a three-dimensional C°° CR structure that cannot be imbedded into C2. Defining a three-dimensional CR structure

is equivalent to specifying a vector field L E TC (R3) so that L andL are linearly independent. A CR mapZ = (Z1, Z2): R3 —p must satisfy LZ1 = 0, LZ2 = 0._We will construct L so that any two functions Z1, Z2: C with LZ1 = LZ2 = 0 must also satisfy dZ1 AdZ2 = 0 at the origin and therefore the map Z = (Z1, Z2): R3 —' C2 cannot be an imbedding of any neighborhood of the origin of The desired vector field L will be a perturbation of the generator of the CR structure for the Heisenberg group. Recall the Heisenberg group in C2 is given

Nirenberg's nonimbeddable example

173

V

S

FIGURE 11.1

by

M = {(z = x+iy,w)

E

C2;y= 1w12}.

The graphing function for M is given by H: JR3 —+

C2

H(x,w) = (x+ilwI2,w). The real tangent space of M at 0 is the copy of JR3 with the coordinates (x, u, v) where w = u + iv. Let C be a circle in the copy of JR2 given by {v = 0}. The circle C is constructed so that the u-coordinate of each point in

C is positive. Each point P0 = (x0, uO, 0) e C is contained in the unique circle

= {(x0, u, v);

u2 +

v2 =

Let

s=uCp. pE C

S is a two-dimensional torus in JR3. We define T to be the open solid torus in R3 whose boundary is S. T is obtained by filling in the circle C with a disc, D, and sweeping it around the torus.

The imbeddabiWy of CR Manifolds

174

With w= u+iv, let C = H{C} = {(x + iIwI2,w); (x,w) E C} CM

S= H{S} = {(x+iIwI2,w); (x,w) E S} CM T

H{T} = {(x + iIwI2,w); (x,w) E T} C M.

Also let 7r1: C2 —' C be the projection ir1 (z, w) = z. Define C1

Si = 7r,{S} CC T1 = 7r,{T} CC.

Note that C, S. and T are circled n the w-variable, which means that if the point (zo, WO) belongs toS (orC or T), then the circle {(ZO, w); wi = IwoI} is also contained in S (or C or T). Therefore, C, is a simple closed curve in C, S1 = C,, and T, is the open set in C whose boundary is C1. LEMMA I

iri{M — is a connected subset of {z E C, Im z O} which contains {Im z = O} (the x-axis). Note that M — T is circled in the w-variable and so ir1 {M — T} = C; Im z O} — T,, which is connected. C does not intersect the x-axis

PROOF

{z

and so neither does C,. Thus, the x-axis is contained in ir1 {M — T}, as desired.

We remark that if 'l is a countable sequence of nonoverlappmg tori constructed as above, then the lemma still holds with T replaced by From Theorem 3 in Section 7.2, the generator for H0" (M) is given by

= Let ir: C2 —'

be

0

—2zw-—

Oz

0

+ —. Ow

the projection given by ir(x + iy, w)

= ir,{L,}

.0 +

(x, w). Define

0

A CR function on M can be viewed either as a function 1: M —' C with L,f 0 on M or as a function 1= 10 H: R3 C with Z1f = 0 on R3. For z E C with Im z 0, let r'(z) be the circle {(z,w);IwI2 = Im z}. So r(z) = fl M.

Nirenberg's nonimbed4able exampk

LEMMA 2

Suppose I: M (a)

If L, I =

C is a C' funerion. 0

on an open set D C M, then the function

F(z)

z

=

f fdw z)

is

(b)

holomorphic on the set {z

C; ['(z) C D}.

E

C M—T then

IfLif_—Oon

f fdw=o.

r(z) (c)

If L,f = 0

on M — 1', then

ffidw Adz =0. For the proof of part (a), we parameterize the circle ['(z) = {(z, w); 2ir. Using Theorem 2 = for 0 by w = w(z,Ø) = in Section 9.1, we can extend f ambiently so that t)f 0 on D C lv!. Here, all we need is that ôf vanish on D c Lvi to first order. We have PROOF

F(z)

Since

=

f fdw

= ôf/thD =

0

on 11) C M, we have

=

=

.

0

foT

+

with F(z) C D.

For (b), if L, f = 0 on M — T, then part (a) implies that F is holomorphic on the interior of the set {z E C; r(z) C M — T}. F is also continuous on the closure of this set. By Lemma 1, this set is connected and contains {im z =

When Im z = 0, F(z) degenerates to a point and so F(z) = F 0 on r(z) C M — T}, as desired.

0.

O}.

Therefore,

176

The Imbeddabilily

of CR

Manifolds

Part (c) will follow from the fact that S is foliated by curves r(z) which z(9),O 0 2ir. satisfy part (b). We parameterize C1 = iri{H(C)} by 0 Therefore, S is parameterized by (0,0) (z(0),w(0,9)) with w(0,0) =

We have 2ir

ff fdw A dz

=

//

J(z(0), w(0,

=7 =

/

z'(O)dO.

Since r(z(0)) C S C M — T, this last integral vanishes by part (b), as desired.

Lemma 2 is valid if T and S are replaced by a nonoverlapping union of such tori. In this case, part (c) is valid for each torus in this union. LEMMA 3

Suppose D is an open set with smooth boundary in M. Let f: D C' function. With w = u + iv Jdw A dz = 2i

where

L1

=

—+

C be a

Jff L, fdxdudv

— 2iw(a/02) is the generator for H°"(M).

PROOF We use Stokes' theorem to obtain

JffdwAdz =fff(df)AdwAdz. We have

dfAdwAdz = = We also have z = x +

on

I of

of Ow

\I AdwAdz.

M and so dz = dx + iwdti) + iiI'dw on M.

177

Nirenberg's nonimbeddable example

Therefore

dfAdwAdz= 4—2iw-'

i9zj

The lemma now follows after rewriting d'i) A dvi as 2idu A dv.

I

Now we construct Nirenberg's nonimbeddable example. Let be a sequence of nonoverlappmg ton in R x C which converge to the origin as i 00. Each 1', is constructed like the torus, T, given at the beginning of this section. Let g: R x C —' R be a nonnegative C°° function that is positive on each The fimction g must vanish to infinite order at the origin. Let L = L1 +g(O/Ox), where L1 = (i9/&th) — iw(O/Ox). The vector field L agrees with L1 to infinite order at the origin. Clearly, L and L are linearly independent (over C) at each point in a neighborhood U of the origin in R x C. Since [L, L] 0, the subbundle L generated by L is involutive. Therefore, (U, L) defines a threedimensional CR structure on U C R x C. This CR structure is not CR equivalent to any three-dimensional CR submanifold of n 1, as the next theorem shows.

THEOREM I Suppose = + g(O/ôx), where g is constructed as above. Suppose Z1, Z2: R x C '—p C are C1 functions with LZ1 = LZ2 = 0 near the origin in R x C.

Then dZ1 A dZ2 = 0 at the origin. In particular, the derivative at the origin of any CR map Z: R x C must have real rank at most 2 and therefore Z

cannot be an imbedding of any neighborhood of the origin of R x C. PROOF L1

Let H: R x C -, M be the graphing function for the Heisenberg (H(x, vi) = (x + iIwI2, vi)). As in the beginning of this section, we let

=

where

Define

M —i R is defined by the equation

o

H =

g.

For a function

TheequationLf=Oona

f:

neighborhood U of R x C is equivalent to the equation Lf = 0 on H{U} C M.

So if Lf =0, then = Since g vanishes on R x C —

we

have L1f =

Therefore

ff fdw A dz = 0

0

on M — {U17}.

The Imbeddrzbilhiy of CR Manifolds

178

by part (c) of Lemma 2. Using Lemma 3, we obtain

o=fffdwAdz = 2ifffLifdxAduAdv =

Since

> 0 on each 7, each of the functions Re(Of/Ox), Im(Of/&r) must

vanish at a point in each T1. Equivalently, each of the functions Re(Of/ax) and Im(Of/Ox) must vanish at a point in T1. Since the converge to {0} as i oc,

Of/Ox must vanish at the origin. From the equations (L1 + g(O/Ox))f = 0 = 0/Oil) we have Of (0)/Oil) = 0. Since both (Of/Oil))(O) = 0 and and (Of/Ox)(0) = 0, we obtain df(O) = Therefore, if Tz1

= 7z2 = 0 near the origin in R x C, then

dZ1(0)AdZ2(O)

(OZ1(0)) (0Z2(0))

This completes the proof of Theorem 1.

I

dwAdw

12 Further Results

In this chapter, we discuss further results, whose proofs are too involved to include in a book of reasonable length. Instead, we refer the reader to the references. We start this chapter with some refinements of the normal form given in section 7.2. Then we discuss the Levi form in more detail. The chapter ends with some facts concerning nongeneric CR submanifolds.

12.1

Bloom—Graham normal form

If the Levi form of M at a point Suppose that M is a real hypersurface in p E M is not identically zero, then the totally real direction of the tangent space

at p can be expressed as the projection of a Lie bracket of the form [L, L e H"°(M). If the Levi form of M at p vanishes identically, then it still may be the case that the totally real direction of the tangent space can be expressed as some higher order Lie bracket generated by vector fields in Hc (M) = H"°(M) H°"(M). This leads us to define the concept of a point of higher type.

To make this concept precise, we need some definitions. For j point p E M, we call an operator of the form

2 and a

Lk EIf'(M) be a Lie bracket of length j at p generated by Hc(M). Let (lvi) spanned (over as the vector subspace of For j 2, we define (M) and all Lie brackets of length k j at p generated by Hc (lvi). C) by typically varies with (M). The dimension of Note that C p and so the union (over p) of these spaces typically cannot be considered as a subbundle of Tc(M). If M is a real hypersurface, then we say that p E lvi

isapointoftypem if

mother and words, a point p E M is a point of type m if the totally real direction of the 179

Fuflher Results

180

tangent space of M at p can be expressed as the projection of a Lie bracket of length m at p generated by Hc (M) but not by any Lie bracket of length less than m. We should mention that there is a different notion of type defined by D'Angelo ED] which is defined in terms of the order of contact of complex analytic varieties.

Now we express the type of a point in terms of coordinates. Suppose the point p is the origin and suppose M is graphed over its tangent space at the origin in the manner described by Theorem 2 in Section 7.2. Coordinates for are given by (z,w) where z = x +iy E C and WE The defining equation for M is given by y = h(x, w) where h : R x R is smooth (say C°°) and where there are no pure terms through order m + I in the Taylor expansion of h at the origin. Let L1,. . , be the local basis for H"°(M) given in Theorem 3 in Section 7.2. If the origin is a point of type m then the (ö/öx)-component of any Lie bracket of the L'8 and the L° of length less than m at 0 vanishes. Together with the fact that there are no pure terms in the expansion of h, we can inductively show that all the w and derivatives of h through order m — at the origin vanish. We have the following expansion of h: .

1

h(x, w) = p(w,

+ e(x, w)

where p is a homogeneous polynomial of degree m in w and with no pure terms and where e is a smooth function that satisfies the estimate Ie(x,w)I

+ IxIIwI + 1x12)

for some uniform constant C > 0. The variable w is the coordinate for H"°(M)

and we assign the weight 1 to w and The variable x is the coordinate for the totally real tangent space direction of M at the origin. Since this direction can be expressed as the projection of a Lie bracket at the origin of length m generated by Hc (M), we assign the weight m to the variable x. Likewise, we also assign the weight m to the normal variable y and to the complex coordinates z = x + iy and = x — iy. The weight of a monomial is by definition the number + + m(j + k). By definition, the weight of a smooth function is the minimal weight of all of the monomials appearing in its formal Taylor expansion about the origin. So the homogeneous polynomial p has weight m whereas e has weight greater than m. Note that in an unweighted sense, the polynomial p may vanish at the origin to higher order than does e. For example, suppose

h(x,w) = IwI2Re{w2} +

1w12x1.

Then the origin is a point of type 4 and the term e(x, w) = IwI2xi has weight 6.

In in unweighted sense, e vanishes at the origin to third order. The above discussion for hypersurfaces generalizes to submanifolds of of codimension d> I. In this case, we follow Bloom and Graham [BG] and say

Bloom-Graham normol form

181

a point p E M has type (mi,.

that

. ,

me,) (with

m3 mk for j < k) if the

following conditions hold:

(M) for j
=

The numbers m1,... , md are also called the HOrmander numbers. Condilion (b) is vacuous if m2 = m2+i. Note that we allow the case where m3 = 00. We say that the point p has finite type if all the m3 are finite. Suppose the point p E M is the origin and suppose the defining equation for

Misy = h(x,w)whereh

issmooth.

:

IfMisrigid(i.e., his

independent of the variable x) then the type of the origin can be determined by examining the order of vanishing (in w) of the components of h. For example, if M = {(zI,z2,w) C3; Xi = lw!2, x2 = IwI2Rew}, then the type of the origin is (2,3). For another example, suppose

M = {(zI,z2,z3,w)

x1 = wi4, x2 = w!2Re{w2}, x3 =

1w16}.

In this case, the type of the origin is (4,4,6).

The nonrigid case is more complicated. We assign weights to the coordinates as follows. As with the hypersurface case, we assign weight 1 to the variables We assume h1,. .. , are ordered so that w and Let h = (hi,. . . (0, w) has the smallest weight among the functions h1 (0, w),.. . , w). to the variable z1 Let be the weight of h1 (0, w). We assign weight (and to and x1, Yi). If this collection of functions vanishes to infinite order, then we assign the weight of oo to z1 (and to yi). Next, we order h2,... , hd so that h2(xi , 0,... , w) has the smallest weight among the functions h2(xi,0,0. . . ,w), . .. ,hd(xI,0,0... ,iv). Let 12 be the weight of h2(xi ,0,O..., w). We assign the weight 12 to the variable z2 (and to x2, 1/2). Continuing in this way, we assign weights (li,. . , to all the coordinates We have (z1,.. . , Unlike the rigid case, the weights of these coming coordinates, in general, are not the same as the numbers (mi... , from the type of the origin. In fact, Bloom and Graham show that if the origin ,

.

is a point of type (mI,...,md) and if the weights (ll,...,ld) are assigned as above, then m1 1 for I i d. Suppose the weights of the z-coordinates are (1k, .. . , id). We say that the manifold M = {y = h(x, w)} is presented in Bloom—Graham normal form if the following three conditions on h are met:

(a) For 1 i d where p' is a homogeneous polynomial of weight is greater than 12.

and the weight of e1

Further Results

182

(b) There are no pure terms in the polynomial p2 (i.e., there are no terms of the form or (c) For 1 j
0

0

=

...

0

(i.e., just

is the differential operator 0/Ow, replace x by 0/Ox etc. in the expression for Pj). where

In requirement (a), p2 is not allowed to depend on the variables xi,... , xd. Requirement (c) means that there are no terms of the form

. . .

tained in p2. For example, if

M = {(zI,z2,W) E C3; lit

1w12, 112 =

xiiwI2}

then the defining functions for M satisfy requirements (a) and (b) but not (c). (because

=

x1p1).

The main theorem in Bloom and Graham's paper is that if a point p M has type (m1,... , with all the m3 finite, then there is a local biholomorphic change of coordinates so that in the new coordinates, the defining equations satisfy Bloom and Graham's normal form with = m8 for 1 i d. Conversely, they show that if M is already presented in their normal form, then the origin

is a point of type (ml,...,md) = (l1,...,ld). Example 1 Suppose M = {(zt,z2,w) E C3; v' = wi2,

1/2

= xiIwj2Rew}. Then M is

presented in Bloom—Graham normal form. The weight of the (zi, z2) coordinates [1 is (2,5) and the type of the origin is also (2,5).

Example 2 C3; 1/i = wi2, 1/2 = xijwI2}. M is not presented in normal form since condition (c) is not satisfied as mentioned above. Since lii = 1w12 on M, we can replace P2 by the expression x1y1 which is the imaginary part of (1/2)z?. We can make the change of variables Li = zi and In the new variables (drop the hat), the defining equations z2 = z2 — for the manifold M become y' = wi2 and 1/2 = 0. Thus, the type of the origin is (2, oo). This example illustrates the importance of condition (c) in the Bloom—Graham normal form. A naive count of the weights of the original defining functions for M might lead one to erroneously think that the type of the origin is (2,4). This example also illustrates some of the ideas used to prove Bloom—Graham's theorem mentioned above. Another illustration is provided in I] the following example.

Suppose M = {(zi,z2,w)

Rigid and semirigid submanifolds

183

Example 3 (See Section 6 in [BGJ.) Suppose

M = {(zI,z2,z3,w) E C;

= wI6, 92 = xiIwl2, y3 =

l'his manifold is not presented in Bloom—Graham normal form because p3 = We note that p3 can be replaced by and furthermore

=

—

= z1 and = z2 and z3 = z3 — (1/3)4. In change variables by setting the new variables (drop the hat), the defining equations for M become We

y2X1 3

Yi

— wi

6

—

92— xiIwI

2

'

93_

3

—

-j--IWI

6

manifold is still not presented in Bloom—Graham normal form because = We apply the same procedure again. We have = After dropping the hat, we see —x1y?). We let z3 = Z3 — that in the new coordinates, the defining equations for M become The

=

= xijwl2,

wi6,

=

The manifold Al is now presented in Bloom—Graham normal form. The type

of the origin is (6.8,24).

12.2

I]

Rigid and semirigid submanifolds

At the end of Section 7.2, we said that a submanifold is rigid if its defining are given equation has the form y = h(w). As before, the coordinates for (here, d is the real codimension by (z, w) where z = x+iy E C and w E of M). The point is that for a rigid submanifold, the graphing function h is independent of the variable x In [BRTJ, Baouendi, Rothschild, and Treves present a coordinate invariant description of rigidity. To describe this condition, we make a few definitions. First, we say that a vector subspace (over C) of the space of smooth sections

of the tangent bundle Tc (Al) to a manifold M is a Lie subalgebra if

is

involutive (i.e., closed under the Lie bracket [ , ] operation). A subalgebra is called abelian if [L1, L2] = 0 for L1, L2 E We say that an abstract CR structure (M, L) is invariant under a transversal Lie group action if there is a

Fwlher Results

184

finite dimensional Lie subalgebra, g, of

(M) with the following properties:

LeLeg =TC(M) [T,G]eE forLEEandGEg. If M is a rigid submanifold of C's, then the vector fields

generate H°" (M) by Theorem 3 in Section 7.2. Since the coefficients of each L, are independent of x, the Lie bracket of each L, with the vector field O/Oxk vanishes. Therefore, if M is a rigid submanifold of C's, then (M, H"°(M)) is invariant under a transversal Lie group action given by the abelian subalgebra generated by linear combinations (over C) of the vector fields

0/Oxi,... ,t9/OXd. In [BRT], Baouendi, Rothschild, and Treves establish the converse: if an abstract CR structure (M, L) is invariant under a transversal abelian Lie group action, then the manifold can be locally imbedded into C" with coordinates (z, w), z = x + iy E Cd where d is the CR codimension of M and where the defining equation for M is given by y = h(w). Another class of CR manifolds of interest is the class of semirigid manifolds. If the CR manifold is imbedded into C" and presented in Bloom-Graham normal form, then we say that M is semirigid if each p, is independent of the variable x. The graphing function h = (h1,. . . for M is allowed to depend on x but the terms of lowest weight in each h1 are required to be independent of x. In [BR2], Baouendi and Rothschild give a coordinate invariant description of semirigidity. To describe it, we need some more For an abstract CR structure (M,L), let X(M) be a complement to L L as explained in Section 8.2. If M is imbedded into C", then X(M) is the totally real tangent

bundle to M. Let X*(M) be the space of one-forms that are dual to X(M). For p E M and 0 E X (M), we let m(p, be the smallest integer m for which there exists a Lie bracket of the form

Lm=[Mi,[M2,...,[Mm_i,Mm]...],]

(1)

0. If no such finite integer m exists then we set m(p, = cc. with (0, As usual, the notation ( , ) denotes the pairing between one-forms and vectors. The abstract CR structure (M, L) is defined to be semirigid at a point p E M if for each 0 x;(M), we have = 0 for all commutators and

of the form (1) withj 2, k 2, and j+k m(p,0). Baouendi

and Rothschild's result states that if an abstract CR structure is semirigid, then

near any point of finite type, it can be locally imbedded into C" so that in its Bloom—Graham normal form, the polynomials

. . .

are independent of

More on the Levi fonn

185

the variable x. They also show that the class of semirigid manifolds contain (1) the class of all real hypersurfaces in C'2; (2) manifolds in which the largest Hönnander number is at most 3; (3) the class of CR structures (M, L) where the complex dimension of L is one and the largest Hörmander number is at most 4; and (4) the class of manifolds where all the Hörmander numbers are the same or, more generally, if the difference between any two Hörmander numbers is at most one.

12.3

More on the Levi form

In Section 10.1, we show that a Levi flat CR manifold (M, L) is locally foliated by complex submanifolds. Furthermore, for each p E M, IL,, is the complex tangent space at p of the leaf of the foliation that passes through p. This result can be strengthened. First, let us extend the Levi form to a bilinear map

by

= where

is the projection p

p

Here, X and Y E L e L are vector field extensions of the vectors X,,, and The Levi form defined in Chapter lOis the restriction of the above map E L,, and X,, = Y,, E to the case where As with the usual Levi form, this definition of

is independent of the vector field extensions

X,Y E LeL. For p E M, let

is called the Levi null set at p. As a simple example, let M = {(z, WI, w2) E C3;Rez = 1w112}. In this case, a basis for L is given by and L1 = — Since the coefficients of L1 are independent of w2, we have [O/0w2, L1] = [0/Ow2, L1] = 0. Therefore, the Levi null set at any

point p E M is spanned (over C) by the vectors O/&W2, 0/0W2. Note that M is foliated by one-dimensional complex manifolds that are translates of the w2-axis.

We can generalize the above example to more general CR structures (M, L). If the dimension of Ni,, is independent of p M, then N = U,, N,, forms a

Further ResulLc

186

subbundle

of L L We have for

(I)

Under this constant rank assumption on N, M can be foliated by complex manifolds such that for each p E M, is the complex tangent space to the leaf passing through p. The proof of this result (due to M. Freeman [Fr]) uses similar arguments to those in the proof of Theorem I in Section 10.1. The key idea is to show that the subbundle N is involutive, i.e., [N, N) C N. To see that N is involutive, let X and Y belong to N and let Z be an element of L L. Then from (1), [X, Z] and [Y, Z] belong to L L. Hence, Z], Y] and Z], X] also belong to L L. From the Jacobi identity YJ,

Z] +

Z], X] = 0

X], Y] +

we conclude that Y]. Z] also belongs to This means [X, Yl belongs to N and so N is mvolutive. It follows that the underlying real subbundle which generates N is involutive and Freeman's result follows from the real Frobenius theorem and the Newlander—Nirenberg theorem.

We should mention that the Frobenius theorem and the Levi form can also be viewed from the differential form point of view rather than the vector field point of view taken earlier in this book. Suppose M is a manifold and L is a subbundle of the real tangent bundle, T(M). Let I C T*(M) be the annihilator of L, i.e.,

1= {th E

(Ø,L) =

0

for all L E

L}.

I is a subbundle of T(M). The condition that L is involutive is equivalent to

the condition that dq) E I A T*(M) for each

E I. This is easily seen from

the equation

AL2) =

—

L2{(Ø,L1)}

—

(see Lemma 3 in Section 1.4). The differential form version of the Frobenius theorem is the following: if I is a k-(real) dimensional subbundle of one-forms with d{I} c IAT(M), then there exist smooth functions u1,... ,Uk such that du1,. . locally generate I over the ring of smooth functions on M. This version follows from Theorem 1 in Section 4.1. — For a CR structure (Au, L), the annihilator of L L is X*(M). The Levi form is designed to measure the degree to which L L fails to be involutive. Therefore from the differential form point of view, the Levi form at p is the map for E 1,, where it,,, is the projection .

it,,

where

:

A2T;c (M)

E I is any extension of

{A2T*c(M)} A

T;C (M)}

Note that if this Levi form vanishes for

Kurantshi's imbedding theorem

all p

187

E M, then d{I} C IA T*C (M) and so L E is involutive. In this case,

M is Levi fiat.

12.4

Kuranishi's imbedding theorem

Nirenberg's example of a nonimbeddable abstract CR manifold shows that the real analytic imbedding theorem in Chapter 11 does not hold, in general, for the CR manifolds. In [Ku], Kuranishi shows that with the additional class of assumption of strict pseudoconvexity and a restriction on the dimension, then The dimension a given C°° CR structure can be locally imbedded into restriction requires that the underlying manifold M have real dimension 2n — with n 4 and that the CR codimension of the CR structure (M, L) is one.

In general, the image of the Levi form at a point p E M (see Definition I in Section 10.1) is a real subspace of If the CR ® e codimension is one, then the complex dimension of ® C/{L,, is one, and therefore the Levi form can be considered a real valued map. A CR structure (M, L) is then said to be strictly pseudoconvex at p if the Levi form is either always positive or always negative. Under these assumptions, Kuranishi's

result states that the CR structure is locally CR equivalent to an imbedded hypersurface in Kuranishi's result only handles CR structures of dimension at least 7. Nirenberg's nonimbeddable example is a strictly pseudoconvex CR

structure of real dimension 3. It is still unknown whether or not a strictly pseudoconvex CR structure of real dimension 5 can be locally imbedded as a real hypersurface in C3.

12.5

Nongeneric and non-CR manifolds

Within the class of imbedded CR manifolds, most of our concern in this book is with the class of generic submanifolds. One reason for this is that "most" submanifolds of are generic. Another reason is that a nongeneric CR submanifold can always be locally presented as the graph of a CR map over a generic CR submanifold of some lower dimensional complex Euclidean space. This is noted in [HT2J. To see this, suppose M is a nongeneric submanifold of Suppose d is the real codimension of M and let k be the CR codimension of M. As explained after the proof of Lemma 1 in Section 7.2, d — k must be a positive even integer which we denote by 2j. Furthermore, there exist coordinates (z, (, w) for where z E such that (= x + iy E Ck, and w E

Al = {(z,(,w)

E

z = H(x,w), y = h(x,w)}.

Further Results

188

x Cn—d+j, where h has Here, h and H are smooth functions defined on values in Rc and H has values in C'. Both h and H along their first derivatives = Ck x Cfl_d+3 vanish at the origin. Let M0 be the submanifold of

defined by

By an easy dimension count, M0 is a generic submanifold of M is the graph over M0 of the map C: M0 '—p M given by

p = (x + ih(x, w), w)

Moreover,

G(p) = (H(x, w), x + ih(x, w), w).

:M The inverse of C is the projection map M0 given by ir(z, (,w) = is a holomorphic map, the restriction of ir to M is a CR map. ((,w). Since Therefore, the graphing map C is also a CR map, as desired. Let us briefly turn our attention to the class of compact CR submanifolds of We mention a result of Wells [W] that states that if M is a compact, oriented, generic, CR submanifold of C't, then the Euler characteristic of M must vanish. For example, the equator of the unit sphere in C" is not a CR submanifold.

In fact in Section 7.1, we show that the equator of the unit

sphere has two points where the real tangent space becomes complex linear (i.e., invariant under the J map). The above-mentioned result of Wells shows that any submanifold of C" that is homeomorphic to the equator of the unit sphere in C" cannot be a CR manifold. Analysis near points where the dimension of the holomorphic tangent space changes is typically messy. For example, the proof of Kuranishi's imbedding theorem involves the solution of the tangential Cauchy—Riemann complex with estimates on a domain whose boundary is homeomorphic to the equator of the unit sphere. The hard work involves obtaining the right estimates near points on the boundary where the dimension of the complex tangent space changes.

To make sense of the notion of CR function and other "CR" objects near points where a submanifold is not CR also requires a lot of work. We refer the reader to the work of Harris [Har] for research along these lines.

Part III The Holomorphic Extension of CR Functions

restricts to a CR funcAs shown in Part II, every holomorphic function on However, not all CR functions are the tion on a CR submanifold Al of restrictions of holomorphic functions. In Part III, we examine geometric conditions on M that guarantee that CR functions on M extend as holomorphic Under certain geometric conditions on functions on some open set ci in M, the open set ci contains an open subset of Al, and under other geometric conditions, ci lies to one side of Al. We are especially interested in CR extension to an open Set that is function independent. That is, we ask: given an open set w in SI, does there exist an open set ci in CTB such that each CR function on extends to a holomorphic function on ci? This is a different question than the one answered by Theorem 1 in Section 9.1, which shows that any real analytic CR function on a real analytic that may CR submanifold of holomorphically extends to an open set in depend on the CR function. If there are no further geometric conditions on the CR submanifold, then CR extension to an open set that is function independent is impossible — even when the CR submanifold is real analytic. For example, let M = {(z,w) e C2; Im z = O}. Each real analytic CR function on an open set C M extends to a holomorphic function on an open set ci C C2 with ci fl M = w. On the other hand, we have W

flue e >0

where

= {(z,w) EC2;(Rez, w)Ew

and urn zI

< r}.

If w is convex, then each

is convex and hence a domain of holomorphy. From the theory of several complex variables, a holomorphic function —' C :

exists that cannot be analytically continued past any part of the boundary of

189

The Holomorphic Extension of CR Functions

190

restriction of f, to w is an example of a real analytic CR function that cannot be analytically extended past any part of the boundary of Since > 0 is arbitrary, we conclude that there does not exist a single open set c C2 to which all CR functions on w holomorphically extend. can In the above example, M is Levi flat. A similar construction of the be carried out for any Levi flat submanifold in This suggests that the Levi form might play a role in CR extension. As we show in the subsequent chapters. the Levi form is a key geometric object that determines whether or in not CR extension to a fixed open set is possible. For a real hypersurface C", we present I-fans Lewy's theorem [LI] which states that if the Levi form at a point is not identically zero, then CR functions holomorphically extend to a fixed open set lying to one side of Al. If the Levi form of Al has eigenvalues of opposite sign, then CR functions holomorphically extend to a fixed open set containing both sides of A!. We then generalize this theorem to the case where M has higher codimension. We present two approaches to the proof of the CR extension theorem since the techniques of the proof are as important as the result. The first approach involves the use of analytic discs, which is an idea pioneered by Lewy and Bishop. The second more recent approach uses the Fourier transform. This technique was first applied to the CR extension problem by Baouendi and Treves. Both techniques are used in today's research problems. The

on

13 An Approximation Theorem

Both the analytic disc and Fourier transform approaches to CR extension currenily use an approximation theorem by Baouendi and Treves. This theorem roughly states that CR functions on a submanifold of can be locally approximated by entire functions on C'. This approximation theorem has an interesting history. There are earlier versions of this theorem that are weaker (and some with errors in their proofs). All of these earlier versions have some convexity assumptions on the Levi form. Then around 1980, Baouendi and Treves caught the CR extension community by surprise. They showed that no convexity assumptions are needed. Moreover, their proof is a simple but clever adaptation of the proof of the classical Weierstrass theorem on approximating continuous by polynomials (see Section 1.1). functions on The statement of the approximation theorem of Baouendi and Treves allows very general systems of partial differential equations. However, we state and prove their theorem for the case of the tangential Cauchy—Riemann equations This version of their theorem has a simpler proof. on a CR submanifold of THEOREM 1

IBTIJ

Suppose p is a point in a generic CR M of of class C2 with dimR M = 2n — d, 0 d n. Given an open neighborhood Wi of p in M, so that each CR function of there exists an open set w2 in M with p E w2 C can be uniformly approximated on W2 by a sequence of entire class C' on functions in C". The proof requires two lemmas which we present after some notation. It will be convenient to choose holomorphic coordinates for C"' (as in Lemma 1 of Section 7.2) so that the given point p is the origin and

M={(z=x+iy,w)EC" x where h: Rd x Let w

= u+iv

Rd

is of class C2 with h(0) =

E C1z_d;t = (x,u)

0

1W',

and Dh(0) = 0. and s = (y,v) E 191

192

An Approximation Theorem

Rd x Rn—d

=

Note that coordinates for C71 can be written as ç = t + is.

Define H: IR'1 x

Rd

x Rn_d by

H(t.v) = (h(x,u+iv),v) (with t = (x,u)). Since h(O) = 0 and Dh(0) = 0, we have H(0) = 0 and aH(0)/Ot = 0. We are only concerned with a small neighborhood of the origin and so we multiply h by a suitable cutoff function and assume that h has support in a neighborhood U1 x U2 in R71 x of the origin. By choosing U1 and U2 small enough, we can arrange for all

(t,v)

Here, IAI denotes the norm of the matrix A, i.e.,

R71 x

Al =

sup A(v)I. IuI=I

By the mean value theorem, we have

IH(ti,v) for all (ti,v),(t27v)

(1)

—

x

Near the origin, M is parameterized by H, i.e.,

M = {t + iH(t, v); (t, v)

U1 x

C R71 x IRn_d}.

M is also foliated by the n-dimensional slices

for v E U2 where

M7,={(t+iH(t,v); t€U1 cR71}. For

(=

let

d(=d(1A...Ad(71. LEMMA I 1 on a neighborhood Uj' with 0 E Uj' CC U1 C R'2, Let g E D(U1) with g C is independent of Im (,for ( E Extend g to C71 so that fl x U2 } be a continuous function. Then for ( E

f(() =

lim

J

C' E

Moreover, this limit is uniform for u

and ( E

n{

x U2 }.

This lemma and its proof should remind the reader of the first part of the proof

of the Weierslrass theorem in Section 1.1 where it is shown that f * with eE(t)

—'

f

193

Even though the integrand in the lemma is an entire function of ( E this lemma alone is not enough to prove the approximation theorem. This is because the domain of integration, depends on the variable ((since (is required to

belong to Me). In the next lemma, we show that if f is CR, then a domain of integration can be found that is independent of ( and the approximation theorem will follow. PROOF

x Rn_d

Define (:

by

—+

((t, v) = t + iH(t, v). For fixed v E U2 C the map t' '—p ((t', v) for t' E U1 parameterizes Pulling back the integral in the lemma via this map, we see that it suffices to show

f(((t,v)) =

f g(t')f(((f', v))e_

lim

det

(t', v)) dt'

t' ER"

where the limit is uniform in (i, v) E U1' x U2. Here, we write t' = (ti, . , E dt' = dt' and (O(/&t')(t', v) is the nxn complex matrix with entries

v) for 1 j, k We replace t' by t —

€s

(for t fixed), and the above integral becomes

f g(t—f8)f(((t—es, v))e_

v)) ds.

det

sER"

(2)

If t E

then g(t — €s)

—i

((t, v) - ((t

1

as

0.

Moreover

- €s, v) =

v)

(€s) +

Therefore, we have

C2[((t, v) — ((t

—

s,

v)]2 =

v) s] + Q(€).

This means that pointwise in s (but uniformly in t E Uj' and V integrand in (2) converges, as

f(((t, v))e

—p 0,

E

(12), the

to det

v))

To show that the corresponding integrals converge, we must dominate the integrand in (2) by an integrable function of s E RTh. Certainly g f c9(/Ot is globally bounded. So it suffices to dominate the exponential term. We have

= c_c2

194

An Approximation Theorem

v) = t + iH(t, v), we obtain

Since

Re{[((t. v) —

((t

v) —

— ES,

H(t

—

v)J2.

From (I), we have Re{[((t,v) — ((t

'

2

—

fs,v)1 }

So the exponential term is dominated from above by which is an mtegrable function of s e By the dominated convergence theorem, the integral in (2) converges (as

—'

0) to

v)) J

v)) ds

det

sEE"

and the limit is uniform in t E Uj' and v E U2. We have over after we show

v) = (O/t9t)(t + iH(t, v)) = I + v). Morev)j < 1/2. Therefore, the proof of the lemma will be complete

f

=1

(3)

sEE"

for all matrices A in the set A—

f n x n complex matrices, A, such that urn Al

IRe Al

and Re A is nonsingular

The integral in (3) is a holomorphic function of the entries in A E A. In addition, if Im A 0 then (3) follows by the change of variables s' AsE and a standard polar coordinate calculation. Therefore, (3) must hold for all matrices

A e A by the identity theorem for holomorphic functions. The proof of the lemma now follows by applying (3) to the matrix A = (t9ç/Ot)(t. v).

I

LEMMA 2

Suppose f is a CR function of class C' on a neighborhood w, of the origin in M. Then, there is a neighborhood w2 in M with 0 E W2 cc w1 C M such that for ( E W2

f

('EMo

Moreover, this limit is uniform in (E w2.

The point of Lemma 2 is that if f is CR, then the domain of integration (M0) is independent of the variable (.

PROOF OF THEOREM I ASSUMING LEMMA 2

Since the integrand is an entire function of ( complete.

E

the proof of Theorem I is

I

PROOF OF LEMMA 2

((U1 x U2} C M, we have

For

t + iH(t, v)

(= for some unique t E U1 and

v

E U2. For fixed v

U2, let

is an (n + 1)-dimensional submanifold of M and its boundary is the union 1}. We will only and the set {((t',Ai');t' and 0 A of be concerned with functions whose t'-support is contained in U1. So the only are M0 and We use contributing components of the boundary of Stokes' theorem to transfer the integral in Lemma 1 over into a sum of an integral over M0 and an integral over ({U1' x U2}. we obtain For ( 2CTL

=

f

+ urn

J

EM,,

The integral over involves the exterior derivative = ôç ± but due to the presence of d(' = dC A.. . A d(7ç, the only contributing term comes from From Theorem 2 in Section 9.1, we may assume that the given CR function is f extended ambiently to so that aj = 0 on w1 C M. Assuming that Ui and (12 in Lemma I are chosen small enough so that ({U1 x U2} C wi, we obtain

J

=

('EM0

+ un

(4)

J

('EM,,

The proof of the lemma will be complete after we show that the second limit on the right is zero. Recall that ( = t+iH(t, v). For (' E there is a unique A with 0 A 1 such that

= ((t',

= t' + iH(t', Ay).

An Approximation Theorem

196

we need to estimate the real part of the exponent.

To estimate We have

Re{[(

It

—

—

—

IH(t,v)

—

H(t',

(5)

From the mean value theorem and the estimate IOH/OtI < 1/2, we have j.H(t, v)

—

jlf(t, v) — JI(t', v)I + j.H(t', v) — I-I(t', Av)I

II(t', Av)I S

C is some uniform constant that depends only on the C'-norm of H. Squaring the above inequality and using the inequality 2ab (a2 + b2) for a, b 0, we obtain where

H(t,v) — H(t', Av)12

—

After noting that tv — AvI < lvi (since O< A

+ 2C2Iv — Av12.

1), the above inequality together

with (5) yields —

(1J2}

—

—

2C2lvl2.

g((') only depends on t' = Re ('. Also, 8çg vanishes on a neighI on Ui). So if (' = t' + iH(t', Ày) belongs to the support of 8g. then lt'I must be bounded away from 0. In addition, (= t + iH(t, v) 0 if and only if both t 0 and v 0. Together with the above inequality, we see that there are constants ri , r2 > 0 so that if ( E Now

borhood of the or gin (since g

with I(I
E

supp 0g. then Re{[( — (I]2}

Therefore

f

C'E

C Al with (I < r2. Here, C is a uniform constant that is independent of €, v, and (. This inequality shows that the second integral on the right side of (4) converges to 0 as f —i 0, provided ( lvi with
for all ( E

For the uniqueness part of the CR extension theorem stated in the next chap-

ter, we need to know that holomorphic functions on a wedge in can be approximated by a sequence of entire functions. By definition, a wedge is an

197

set of the form w + {F fl B6} where w C M is an open neighborhood of (the space of vectors at p a given point p E M, where r' is a cone in of radius is the ball in and where which are orthogonal to open

c centered at p. The proofs of Lemmas I and 2 can easily be modified to prove the following approximation result. THEOREM 2 > 0 be given. Let r be Let w1 be a neighborhood of a point p E M and let M). There exist > 0 and a neighborhood w2 of p in M with a cone in ( + {I' fl } and continuous up w2 C WI such that F is holomorphic on of a sequence of entire to Wj, then F is the uniform limit on w2 + {1' fl functions.

14 The Statement of the CR Extension Theorem

We start this chapter by stating Hans Lewy's CR extension theorem for hypersurf aces. The generalization of this theorem to CR submanifolds of higher codimension is then given. This is followed by a number of examples. The proofs of these theorems will follow in Chapters 15 and 16.

14.1

Lewy's CR extension theorem for hypersurfaces

Let M = {z E C'2; p(z) = 0} be a hypersurface in C'2 where p: C'2 —' R is smooth with dp 0 on M. If p is scaled so that IVp(p)I = 1, then from Theorem I in Section 10.2, the Levi form of M at p is the map

w

Vp(p)

for W =

E

When we speak of the eigenvalues of the Levi form of M at p. we are referring to the eigenvalues of the matrix that represents the Levi form (i.e., the restriction of ) (p)) to Let = {z E C'1;p(z) > 0} and = {z E C'2;p(z) <0}. Hans Lewy's theorem can now be stated. THEOREM I HANS LEWY'S CR EXTENSION THEOREM FOR HYPERSURFACES (Li)

Let M be a real hypersurface in C'2,n

2 of

class Cc,3 < k

oc,

and let p

beapointin M. (a)

198

If the Levi form of M at p has at least one positive eigenvalue then for each open set w in M with p E w, there is an open set U in C'2 with p E U such that for each CR function f of class C' on w, there is a

Lewy's CR extension theorem for hypersurfaces

199

unique function F which is holomorphic on U fl and continuous on U fl such that FIUnM = If the Levi form of M at p has at least one negative eigenvalue, then the conclusion of part (a) holds with replaced by If the Levi form of M at p has eigenvalues of opposite sign, then for each open set w in Al with p E w, there is an open set U in with p E U such that each CR function of class C' on w is the restriction (on U fl w) of a unique holomorphic function defined on U.

I

(b)

(c)

Parts (a) and (b) describe one-sided CR extension results whereas part (c) describes a two-sided CR extension result. Note that the quantifiers are arranged so that the open set U depends only on w and not on the CR function defined there. It is easier to see the geomethc meaning of this theorem by choosing coor-

dinates (as in Lemma 1 of Section 7.2) so that the point p is the origin and so that the defining function p has the form p(x + iy, w) = y — h(x, w) for z = x + iy C and w E From Corollary I in Section 10.2, the Levi form can be identified with the map

W = (w,,.

. .

i—i

82h(0)

Since we can arrange coordinates so that all the second-order pure terms in the

expansion of h (about 0) vanish, the Levi form of M describes the secondorder concavity of M near the origin. A positive eigenvalue of the Levi form indicates that M is locally concave up (here, up means toward positive y) along a direction in (i.e., in one of the w-directions). In this case, Lewy's theorem states that CR functions holomorphically extend above M (i.e., to {y > h(x,w)}). A negative eigenvalue of the Levi form means that M is locally concave down along one of the w-directions. In this case, CR functions holomorphically extend below M. If the Levi form has eigenvalues of opposite sign, then the origin is a saddle point for M and CR extension to both sides of M is possible. Since holomorphic functions are real analytic, part (c) of Theorem 1 implies the following regularity result for CR functions. THEOREM 2

of class Suppose M is a hypersurface in (3 k oo) and suppose p is a point in M where the Levi form has eigenvalues of opposite sign. Then each CR function on M that is a priori C' in a neighborhood of p must be of class in a neighborhood of p. If in addition M is real analytic, then an a priori C' CR function defined near p must be real analytic near p.

The Statement of the CR Extension Theorem

200

Theorems 1 and 2 are also valid for CR distributions. This will be briefly discussed in Chapter 17.

14.2

The CR extension theorem for higher codimenslon

The Levi form is also the key geometric object that governs CR extension from a CR submanifold of higher codimension. If M is a generic CR submanifold of real dimension 2n — d, I d n — 1, then the normal space of M at a point p E M (denoted is isomorphic to Rd with p as the origin in

We consider the extrinsic Levi form at p. 4: —i The definition of 4, along with a coordinate description of 4 in tenns of an appropriate system of defining functions, is given in Section 10.2. As with the hypersurface case, the image of the Levi form at p provides information about the second-order concavity of M near p. For p M, let

=

{the convex hull of the image of

C

is a cone, i.e., if v is an element F,,,, then Ày also belongs to for allA 0. If M is a real hypersurface, then R and is either {0} (if 0) or a ray (if 4 is positive or negative semidefinite) or all of R (if C,, has eigenvalues of opposite sign). The translation of Lewy's hypersurface theorem

into these terms is the following: if F,, is a ray, then CR extension is possible to one side of M; if F,,, is all of N,,(M) R then CR extension is possible to both sides of M. If d = codimaM is greater than one, then 4 is vector valued and so is more complicated. As we shall see in Theorem I below, F,, determines the shape and size of the open set to which CR functions holomorphically extend. To state the theorem, we need some additional notation. For two cones

and F2 in N,,(M) we say that r'1 is smaller than F2 (and write F1 < F2) if F1 fl S,, is a compact subset of the interior of {r2} fl S,,, where S,, is the unit sphere in N,,(M). For example, if the codimension of M is two, then N,,(M)

is a copy of R2. In this case, if F1 and F2 are convex cones with F1 < F2, then either F1 = F2 = N,,(M) or else F1 c F2 and the angle formed by the boundary rays for F1 is smaller than the corresponding angle for F2. Note that N,, (M) is always a smaller cone than itself. For denote the open ball in N,,(M) centered at the origin = p of 0, let

radiusf. FortwosetsAandBinC",weletA+B={a+b;aEAandbE B}. THEOREM 1

CR EXTENSION FOR HIGHER CODIMENSION (BPJ

k oo) with Let p be a poim in M so that F,, has

Suppose M is a generic, CR subman(fold of

dimp M = 2n

—

d,

1 d n—

1.

of class

The CR extension theorem for higher codiniension

'7,

17,

F1(;URE 14.1

respect to nonemptv interior ( ) j. Then fyi eveiv in .'t I and an open set !? in p in .1!. there is an opeii set (a)

p

E

(b) in

C

Ti

iu Ii that

:ti c

ope,i ColiC I < Fe,. f/ieee and an E > U so that

± (c)

of

toe cue/i CR Juintion

.1

qt c/usc (

function F defined on

a C(?nhiC'(tC'd li('i,'/iboIh()o(l

ot p

SB,) 5 .11 r/u're is a unique /10/uon uit/i 1and continuous on .1

on

case, the set depends only on and and not Part (b) of the theorem conveys the following on the CR function defined on = 2. picture of which we draw in the case il In Figure 14.1, the picture on the right is a side view with \i going into the fl B. is represented by the shaded region. The quantifiers imply that page. The closer F'1 gets to Fr,. the smaller and and F depend on the cone usually get as shown by the examples in the next section. By allowing and to approach we see that the tangent cone of at p is spanned by the real tangent space of .1/ at p. .11) because In Theorem 1, if V1,(\I ). then we may let fl contains an open set H, .V,,( .\I) is smaller than itself. In this case. As with the

The Statement of the CR Extension Theorem

202

Therefore, if = then which is an open neighborhood of p in each CR function near p is locally the restriction of a holomorphic function defined in a neighborhood of p. This is analogous to the two-sided CR extension result (part c) of Lewy's theorem for a hypersurface. THEOREM 2

oc, and Suppose M is a generic CR submanifold of of class CRC, 4 k suppose p is a point in M with = Then for each neighborhood w of p in M, there is an open set in with p E such that each CR function which is of class C' on is the restriction of a unique holomorphic function defined on Il.

Since holomorphic functions are real analytic and hence C°°, Theorem 2 implies the following regularity result for CR functions. THEOREM 3

(4 k oo) and Suppose M is a generic CR submanifold of of class = suppose p is a point in M with Then each CR function that is a priori C' in a neighborhood of p in M must be of class in a neighborhood of p. If in addition M is real analytic, then each CR function thai is a priori C' in a neighborhood of p must be real analytic in a neighborhood of p. Theorems 1, 2, and 3 hold for CR distributions as well as C'-CR functions. This will be discussed briefly at the end of Part III. is nonempty imIn Theorem I, the hypothesis that the interior of in poses some restrictions on the codimension of M. For example, if dimc

=

then the image of the Levi form is contained in a one (real) dimensional = 1, subspace of Therefore if codimRM 2 and if dime then the hypothesis of Theorems 1, 2, or 3 is never satisfied. By using the bilinearity and conjugate symmetry of the Levi form, it is an easy exercise to in show that if the interior of is nonempty, then m(m + 1) 2d where m = n—d = dime H"°(M) and d = codimRM. The reader should not get the impression that CR extension is impossible if m(m + 1) <2d. However, 1

CR extension in this case requires an analysis of higher order commutators from He (M). Results along these lines will be mentioned at the end of Part III.

14.3

Examples

As already mentioned, the above CR extension theorem for a real hypersurface

is simple to interpret. If the theorem applies, then CR functions either locally

Examples

203

extend to one or both sides of the hypersurface depending on the sign of the eigenvalues of the Levi form. The next easiest class of submanifolds to analyze is the class of quadric submanifolds of C with real codimension two. In Section 7.3, we established that any such quadnc is biholomorphic to one of the following four normal forms. Here, the coordinates for C are given by (21, 22, WI, W2) with 23 = X3 +27)3, j = 1,2. (i)

M

= q(w. th),

7)2 = O} where q: C2 x C2 —. C is a scalar-valued

quadratic form (ii) M={yi=1w112,

(iii) M = {Yi = w112, (iv) M = {YI =

= 7)2 =

Analyzing the CR extension phenomenon for each of these four examples

will characterize the local CR extension phenomenon on all codimension two

quadrics in C. Recall that the Levi form at the origin of the quadric {y =

can be

identified with the vector valued quadratic form w s—p q(w, ti)).

Example 1 Suppose M = {yj = 0 (the origin). The image 7)2 = O}, and p of the Levi form (w '-÷ (q(w, 11)), 0)) is contained in the line {Qji 0); Yi E IR}. In this case, the interior of r0 with respect to No(M) is empty and so the CR extension theorem does not apply. In fact, CR extension to a function independent open set is impossible because M is contained in where ,

each

=

<€} is a domain of holomorphy.

U

Example 2

The image of the Levi form (wl,w2) 0} which = ['o = {yi is already convex. The CR extension theorem states that CR functions on a neighborhood of the origin in M extend to holomorphic functions on an open set fl in C" whose normal cross section (at the origin) contains sets of the type where r, is any smaller subcone than {yI ? 0) in No(M) and where 0 depends on I's. Note that M is the intersection of the convex boundaries = 1w112} and {Y2 = 1w212}. Therefore, CR functions on M cannot holomorphically extend past {y, w,12, 1w212}. In particular, this shows that in general, we cannot take = F0 in the conclusion of the CR extension theorem (because in this example, BE n F0 is not contained in > This example also shows that in general, must {Yi depend on the cone r,; for if 0 is fixed, then n F, is not contained (1w112,1w212) is the quadrant

The Statement of the CR Extension Theorem

204

{ Yi

Wi

2

•1

y1

FIGURE 14.2

in {y' O,Y2?O}.

Y2 1w212} provided

lI'1

is sufficiently

close to r0 =

LI

Example 3 The image of the Levi form A'! = {y' = 1w112, Y2 = £o(wi,w2) = (1w112, Re(wiü)2)) is the half space {y1 > O} together with the Suppose

origin. The image of C0 is convex but not closed. The CR extension theorem states that CR functions on an open subset of the origin holomorphically extend to an open set in C4 whose normal cross section at the origin contains sets of the type Bf fl F1 where F1 is any smaller cone than {Yi > O}. The tangent 0 > O}. cone of at the origin contains the half space Example 4

Suppose Al = {yj = Re(wiü)2). Y2 = Im(w1th2)}. The image of the Levi This can be is all of No(M) form £o(w1, w2) (Re(with2), seen by setting w2 = I and letting w1 range over the complex numbers. Therefore, Theorems 2 and 3 from the previous section apply to this example. A CR function in a neighborhood of the origin in It'! is the restriction of a holomorphic function defined on a neighborhood of the origin in C4. In addition, an a priori C'-CR function near the origin must be real analytic near the origin. LI

205

Examples

In the above four examples, the image of the Levi form is convex. This is not always the case for submanifolds of C" for n > 4 as illustrated by the following example.

Example 5 Let (z1, z2, Z3, Z4, w1, w2) be the coordinates for C6. Let

M={Im zi=1w112, Im z2=1w212, Im Z3=

Im z4=

Here, M has codimension four in C6. The image of the Levi form at the origin

isthe cone

{YER4;yiO,y2O and This set is not convex and it has no interior in R4. However, its convex hull is

the set

{yER4;yiO,y2O and interior in R4, and so the CR extension theorem

which does have

applies to this example.

U

15 The Analytic Disc Technique

In this chapter, we present the proof of the CR extension theorem using the technique of analytic discs. The rough idea is the following. In Chapter 13, we showed (without any assumption on the Levi form) that a CR function on a CR submanifold M can be uniformly approximated on an open set w C M by a sequence of entire functions. To extend a given CR function to an open set in it is natural to try to show that this approximating sequence of entire functions is uniformly convergent on the compact subsets of Il. This can be accomplished by the use of analytic discs. Let D be the unit disc in C. An analytic disc is a continuous map A: D —÷ which is holomorphic on D. The boundary of the analytic disc A is by definition the restriction of A to the unit circle S' = OD. Often in the literature, the analytic disc and its boundary are identified with their images in Suppose that } is a sequence of entire functions that is uniformly convergent to a given CR function f on the open set w C M. Let us say we wish to show that {F3} also converges on an open set The idea behind analytic discs is to C show that each point in is contained in (the image of) an analytic disc whose boundary image is contained in w. From the maximum principle for analytic functions, the sequence of entire functions {F3 } must also converge uniformly

on ft So our CR extension theorem is reduced to a theorem about analytic discs, which we state in Section 15.1. In Section 15.2, this analytic disc theorem is established for hypersurfaces. The proof for hypersurfaces involves an easy slicing argument and thus we obtain an easy proof of Hans Lewy's original CR extension theorem. In Section 15.3. we prove the analytic disc theorem for

quadric submanifolds. The proof here is harder than for hypersurfaces but it is still relatively easy since the analytic discs can be explicitly described. The construction of analytic discs for the general case requires the solution of a nonlinear integral equation (Bishop's equation). This is discussed in Section 15.4.

In Section 15.5, we complete the proof of the analytic disc theorem for the general case.

206

Reduction to analytic discs

15.1

207

Reduction to analytic discs

The key result concerning analytic discs is the following. THEOREM 1

ANALYTIC DISCS

of class Ck, 4 k $ oc with n—i. Letpbeapoint inMsuchthatthe interior

Suppose M is a generic CR subman:fold of

dimaM=2n—d, I

is nonempry. Then for each neighborhood w of p in M and for of in each cone 1' < there is a neighborhood wr C w and a positive nwnber such that each point in WF + {1' fl B(r } is contained in the image of an analytic disc whose boundary image is contained in W. PROOF OF THE CR EXTENSION THEOREM FROM THE ANALYTIC DISC THEOREM

Suppose p e w C M is the given point in the CR extension theorem and let f be a CR function on the open set w. By Theorem 1 in Chapter 13, there is a sequence of entire functions F3, j = 1,2,... which converges to f on some open set W2 with p w2 C w C M. Now we apply the analytic disc theorem with ci.'

Let

fi

U

r 0 be given; there exist > 0 and a neighborhood W2 of p in M with W2 C such that if F is holomorphic

+ {r1 n Bj and continuous up to w1, then F is the uniform limit on fl of a sequence of entire functions n = 1,2 + By the identity theorem for holomorphic functions, it suffices to show the following: suppose F is holomorphic on + {F1 fl and continuous up to w1; if F = 0 on Wj, then F 0 on an open subset of wi + {I'1 fl So we can assume that the approximating sequence from the previous paragraph converges uniformly to zero on W2. From Theorem 1, it follows that there is such that each point in U is contained an open subset U of w2 + {ri fl in the image of an analytic disc whose boundary image is contained in w2. The maximum principle implies that converges to zero at each point in U. Therefore, F 0 on U, as desired. I

The Analytic Disc Technique

208

The proof of the uniqueness part of Theorem 2 in Section 14.2 is easier. Here, the open set contains an open subset of M. Therefore, uniqueness follows from Lemma 2 in Section 15.1.

15.2

Analytic discs for hypersurfaces

In this section, we prove the analytic disc theorem (and hence Lewy's CR extension theorem) for hypersurfaces. The proof is particularly simple in this case since we can obtain the analytic discs by an elementary slicing argument. Using Theorem 2 in Section 7.2, we can arrange coordinates so that the given point p E M is the origin and

M={(z=x+iy,w) where h: IR x —i R is of class C3 with no pure terms in its Taylor expansion through order 2. From a Taylor expansion of h about the origin, we have

h(x, w) =

qjkwjwk + 0(3)

n—d is the matrix for the Levi form = 1 5 j, of M at the origin. Here, 0(3) denotes terms depending on both w and x which vanish to third order at the origin. Since Q = is a Hermitian symmetric matrix, the w coordinates for can be chosen so that Qis diagonalized. This is accomplished by finding a unitary matrix U so that tuQu is diagonal and then letting ii) = U w. The hypersurface A! divides a neighborhood of the origin in into two where

sets

y>h(x,w)} y 0. From the Taylor expansion of h, we have

h(0,w1,0) = qiiIwiI2 + 0(Iwi Is). Let w be an open subset of M which contains the origin. Since

is positive, any small translate of the complex line {(O, w1, 0); w1 E C} in the positive y direction will intersect the open set in a simply connected open subset of this translated complex line whose boundary is contained in By continuity,

209

Analytic discs for hypersurfaces

y

.

.0

'WI'

FIGURE 15.1

the same can be said for small translations of this complex line in the x an directions. More precisely, there are positive numbers 6, > 0 w2 <6, then the complex line such that if < and si.. 1w2k... , —

= {(x + intersects

denote by

,

in a simply connected open subset of whose boundary lies in w.

( E C} which we

The union of the

< 6 contains an open subset of çlt From the Riemann mapping theorem, each is biholomorphic to the unit disc in C. This biholomorphism provides the desired analytic for yJ <

and

1W21,

disc, and the proof of the analytic disc theorem for the case Fo = {y 0} is complete. If instead,

<0 (i.e., F0 =

0}), then the same arguments can be used which is foliated by the images of analytic to construct an open set Il— C has eigenvalues discs whose boundaries are contained in w. If the of opposite sign, then the union of together with fl M forms a set which contains an open neighborhood of the origin and which is foliated in by analytic discs whose boundaries are contained in w. This completes the proof of the analytic disc theorem for the case of a real hypersurface in The proof of Lewy's CR extension theorem for hypersurfaces does not require the Riemanu mapping theorem. Instead, one may use the maximum principle

The Analytic Disc Technique

210

directly to each - to conclude that the approximating sequence of entire functions converges on Q+ (or 11 or There are other proofs of Hans Lewy's theorem which do not use the approximation theorem. Indeed, Lewy did not have the approximation theorem at his disposal. Previous proofs of Lewy's CR extension theorem still arrange coordinates so that geometry of M looks like the picture in Figure 15.1. From there, two approaches can be used; one involves the Cauchy integral formula on each - the other involves the solution of a 0-problem on the complex

Hörmander's proof in his book [Ho]). The analytic disc proof presented here more easily generalizes to higher codimension. lines

15.3

(see

Analytic discs for quadric submanifolds

The slicing argument used in the case of a hypersurface won't work for submanifolds of higher codimension, for if codimR M> 1, then complex lines usually

have empty intersection with M. However, if M is a quadric, then the proof of the analytic disc theorem is still relatively easy because the required analytic discs can be explicitly described. From Section 7.3, a quadric submanifold M is given by

M={(z=x+iy,w)ECdxCY4_d; y=q(w,th)} where q: Cn_d x Cd is a quadratic form. We assume the point p E M given in the analytic disc theorem is the origin. There is no loss in assuming this because a neighborhood of any point p E M can be biholomorphicaily mapped to a neighborhood of the origin by using the group structure on M (see Section 7.3). We wish to fill out an open set with images of analytic discs whose boundaries are contained in a given open neighborhood of the origin in M. Let us start We want to find an analytic disc with a given analytic disc W: D D) so that the analytic disc given by A(() = (G((), C: D —÷ C'1 has boundary image in M. We will then show that by varying W, the images of the corresponding analytic discs, A, will fill out the desired open set ft is given by a convergent power series The analytic disc W: D —÷ a,€Cn_d, In our application, all but a finite number of the parameters {ao, a1,.

=

vanish. In order for the set {A(() = Cd must satisfy in M, the analytic disc C: D

Im G(() =

for

KI

1}

1.

.

.} will

to be contained

Analytic discs for quadric

After substituting W(() = E and expanding the right side using the bilinearity and symmetry of q, this equation becomes

Im G(() =

1)

Ok<j

jrrO

The

=

(since

q

term within the brackets on the right is analytic in (

E

D and so the analytic

disc

= x E Rd) satisfies the equation Im G(() = for 1(1 = 1. Note that = 0) = x + i q(a,, a3). We summarize this discussion in the following lemma. LEMMA 1

Suppose {ao,ai, . . .} is a given sequence of vectors in co. Let x given by

be given. The analytic disc A:

—'

with

A(() =

a3

2

W(c))

W(()

= G(ç) j=O

satisfies

for

A(ç=O)=

(j=l ao).

The vectors x E and ao, a .,... E Cn_d are parameters at our disposal. Each parameter generates an analytic disc A with boundary image in M. Now we will show that by varying these parameters, the images of these discs will sweep Out the desired open set 11 for the analytic disc theorem. We consider the center of the disc, = 0). It will be convenient to let a0 = w E

The Analytic Disc Technique

212

From Lemma I, we have

A(ç=O)

(1)

x Here, we have written the coordinates of a point in as (x, y, w) E Rd x The first term on the right describes a point in M. We claim the second term on the right lies in the closure of the convex hull of the image of the Levi form. To see this, recall that the Levi form of the quadric submanifold M at the origin is the map y = q(w, tD) for w E The convex hull of the image of q is the cone

N

ro =

1, 0

t3

1,

=

1 and

aj E

t 0, we have

Since

a3 e Cn_d and a3 =0 for all but a finite number

Equation (1) now shows that the set of all centers {Á(ç = 0)}, where A is given as in Lemma I, contains the set M + I'o. The

proof of the analytic disc theorem requires a local version of the above

analysis. The parameters x, w, and a1, a2,... must be restricted so that the

boundary of the corresponding analytic disc A is contained in the open subset w C M given in the analytic disc theorem. In return, we must settle for sweeping nF} out an open set which contains subsets of M + r'0 of the form 'wr + for F < Fo. For vectors X ,.. , X E N0 (M) Rd, let •

= the convex cone generated by X1

If F < f'o is the cone given in the analytic disc theorem, then X1

. . ,

can

be chosen in the image of the Levi form (q) so that

XNcFO. For

each j = 1,..., N, there is a vector a3 E

with

X3 =q(cxj.äj).

(t, ER) forl <j N. From Lemma 1, we obtain a family of analytic discs A(t,x,w): D —'

Analytic discs for quadric subman(foids

213

which depend continuously on the parameters t = (t1,... , tN) From the formula for A, we have x E Rd. e

and

IA(t. x,

<

C is a uniform constant that is independent of the parameters t, x, w and the variable ç E D. Suppose w is the open neighborhood of the origin in M given in the analytic disc theorem. From the above inequality, there is a w) > 0 so that if (ti, 'xI, Iwl
obtain

A(t,x,w)(( = 0) = (x,q(w,th),w) +

(2)

< 6} is an open neighborhood of the = {(x,q(w,ü)),w); The set origin in M. Inaddition, the set {E3_1 = I(t,,...,tN)! e(b) > 0. Therefore, the set for some suitably small the set ['XI...XN fl

{A(t, x, w)(( = 0);

<

fl F} (since F C Fxl,..xN), and moreover, each contains the set wF + is contained in the image of an analytic disc (namely A(t, x, w)) point in whose boundary is contained in w. This completes the proof of the analytic disc theorem (and hence the CR extension theorem) for the case of a quadric submanif old.

The discussion immediately following (1) suggests that the analytic disc theorem for quadric submanifolds has a global version. That is, we showed that the set M + F0 can be realized as the union of centers of analytic discs whose boundary images lie in M. So it is not surprising that the CR extension theorem for quadrics has a global version as well. THEOREM I Suppose M is a quadric submanifold of C11. If the interior of { Fo } is nonempty, then for each CR function f that is of class C' on M, there is a function F

that is holomorphic on

= M + interior {Fo} and continuous on Il U M with

= f. In general, there is no global version of the approximation theorem. Therefore, we use the local version of the CR extension theorem together with the group structure on M to prove Theorem 1. Let F be any smaller subcone PROOF

of F0. From the local CR extension theorem, there is an open set w in M containing the origin and an c > 0 such that the given CR function f on M extends holomorphically to Wj- + {F fl

(zj,vji)o(z2,w2)=(z,

The group structure on M is

+z2+22q(wl,w2),wI +w2)

214

The Analytic Disc Technique

for (z1, w1), (z2, w2) E M. Note that the group structure preserves M + r for any convex cone r C No(M). Let p be any point on M. The map gp(Z,W) = (z,w) op is a holomorphic map which takes wr + onto the open set

letting p range over Al, we see that the CR function / extends to a holomorphic function on M+{rflBEF}. This argument }. By

can be applied to any translate of M lying in M + {I' fl }. Continuing in this way, / extends holomorphically to M + r. The proof of this theorem is I completed by applying this argument to each smaller subcone 1'
15.4

Bishop's equation

For a quadric submanifold Al = {(z = x + iy, w); y = q(w, iii)}, the construction of the analytic discs is easy. Given an analytic disc W: an explicit formula is given for the analytic disc G: D Cd (see Lemma i in the previous section) so that the boundary of the analytic disc A = (G, W): D C'2 is contained in M. In this case, the explicit formula for C is possible because the graphing function for M (namely q) is independent of the variable x. However, the graphing function of a more general CR submanifold of C'2 depends on both x and w and the corresponding construction of analytic discs requires the solution of a nonlinear integral equation (Bishop's equation [Bi]) which we now discuss. As usual, we may assume that coordinates have been chosen so that the given point p e M is the origin and

where h: Rd x is smooth (class C4) and h(O) = 0, Dh(0) = Given an analytic disc W: D 0. we wish to find an analytic disc C: D —p Cd so that the boundary of the disc A = (G,W): D C'2 is contained in M. This means that G must satisfy

ImG(() = h(Re

for

=

1.

This equation involves both u = Re C and v = Im G, whereas in the quadric case, the corresponding equation only involves v = Im C. The above equation

will be easier to solve by eliminating either u or v. To do this, we use the Hilbert transform which is defined as follows. Let S' be the unit circle in C. If w S' —. Rd is a smooth function, then u extends to a unique harmonic function on the unit disc D. This harmonic function has a unique harmonic conjugate in D (denoted by v) which vanishes at the origin. The Hubert transform of u (denoted Tu: S' W') is defined to be If C = u+iv: D —' Cd is analytic and continuous up to S', then T(ulsi) vlsi + C where C = —v(( = 0). The function —iG = v — iu is also analytic and so = —u + x where

215

Bishop's equation

x=

Rd are continuous functions with 0). Conversely, if u, v: S' C" is the boundary values of a unique analytic u = —Tv+x, then u+iv: 51

disc G:

C"

with ReG(( =0) = x.

= Suppose u + iv = C: D —+ C" is an analytic disc with 2ir. We apply —T to both sides of this equation and W(e'4')) for 0 5 obtain

= —T(h(u,

0

+ x,

(I)

where x E Rd is the value of u at ( = 0. The above equation will be referred to as Bishop's equation. Conversely, suppose the analytic disc W: D Rd is a solution to and the vector x E Rd are given, and suppose U: S' Bishop's equation. From the above discussion, the function

+ is the boundary values of a unique analytic disc C: C'. Since Re = is contained W): D the boundary of the analytic disc A = (C, in M. Furthermore, Re C(( = 0) = x. We summarize this discussion in the following lemma. LEMMA I

is an analytic disc and x E Rd. If u:

Suppose W: D

Rd

is a

continuous function that satisfies Bishop's equation

<2ir

+x

u(e'4') = then there is a unique analytic disc G:

that satisfies

(a) ReC((=O)=x = u(e141), OS

(b)

Re

(c)

the boundary

5 2ir

of the analytic disc A = (C, W):

is contained

in M. Now, we discuss the solution to Bishop's equation. This requires no convexity assumption on the Levi form. In the next section, we shall use the convexity assumption on the Levi form to choose the right family of analytic discs so that the associated family of discs A = (G, W): D —' Ctm W: D (from Lemma 1) sweeps out the desired open set fl for the analytic disc theorem. To solve Bishop's equation, we must set up certain Banach spaces. The Hubert transform is not a continuous linear map on the space of continuous functions on S' with the sup-norm. Instead, we consider the space of Holder continuous functions, which is defined as follows. Suppose L is any normed linear space (usually either R" or C"). A continuous function f: S' —* L is said to be HOlder continuous with exponent (0 < 5 1)11 there is a finite positive The for 0 5 number M so that 5 Mfr/1 — 5

The Analytic Disc TechnIque

216

set of all such functions is denoted by C°(S', L). If the space L is unimportant for the discussion at hand, then it will be omitted from the notation. The space C°(S') is a Banach space under the norm —

sup

Aa

A

the usual sup-norm. The Hilbert transform is a continuous map from Ca(SI,Rd) to itself. To show this, we need an integral formula for Tu. It suffices to consider the case d = 1. If u: 51 —. R is a continuous function, then its harmonic extension is given by Poisson's integral formula where

is

=

—

7 2irJ

o < r < 1, —

<

<

—

—

0

Letting

z=

rezO

and ( =

this becomes 1

1 fl—IzP

J IC 1= 1

=

f 1(1=1

Since the quantity inside the brackets is holomorphic in z for IzI < 1, a harmonic

conjugate of u is given by

u is real valued. Note that v(O) = Im{(l/2ir) Therefore, Tu is the boundary values of v on S'. 1. Therefore, the The function + z)/( is smooth for (I = 1 and Izi will follow from the continuity of the continuity of T: ca(s1) where Ku is the boundary values on Cauchy kernel K: —, S' of the function 1

1

J

KI=I

(—z

zI
217

Bishop's equation

LEMMA 2

Let 0<

< 1. Then

(a)

K: C°(S')

—

(b)

T:

—

is a continuous linear map. is a continuous linear map.

This lemma holds for a wide class of singular integral operators which includes both K and T. However, the special properties of the Cauchy kernel can be used to give an easy proof of the lemma for K (and hence for T). (51), we first extend u to a PROOF To prove the continuity of K on R so that function E(n): D is defined the same way as C that is independent of u. Here, (S') except that the domain of definition is D rather than S'. To construct

E(u). we extend u to be constant on any line that is normal to S1, and then we multiply this function by a suitable cutoff function. Therefore to prove part (a) it suffices to show K: is continuous. —+ First, we estimate For z E D, Cauchy's integral formula yields

K

-

Ilulic, +

I > 0, we have 1, where Cc, for Izl — is some finite positive constant that is independent of z. It follows that

Since

Cc,IIuIIc, + Next, we estimate Ku(z1) — Ku(z2) for z1, Z2 formula, we have

K(u)(z1) — K(u)(z2) = 1

f

K =

—

D. By Cauchy's integral

The Analytic Disc Technique

218

Let f = Izi — z21 and

{(

K(u)(z1) - K(u)(z2) =

E

C; K

<2€}. We rewrite this equation

— zil

u(ZI))

J (ES'

-

(U

u(z2))

d(

flB2e

2irz

f ,j

u(z2)_u(zl)d( (—z2

(ES'B2e

f

+

]

—

(ES'—B21

(u(() — u(zi))d(+u(zi)—u(z2).

By Cauchy's integral formula, the second integral on the right is

f

u(zl)_u(z2)d(

(ES'

Substituting this for the second integral and simplifying, we have

K(u)(zi) — K(u)(z2)

=

1

v(zi))

J

U(z2))

—

=

—

d(

CE S'

jf

2irz

u(zl)_u(z2)d(

CE 5'

f

2iri

J

—

Z2)(U(()

—

((—z1)((—z2)

CES'—B2,

Let A1, A2, A3 be the first, second, and third integrals on the right, respec-

tively. For A1, we have 1A11

<

J

(K -

+K-

CE S'

Parameterizing

the unit circle by (= (zi/IziI)e*t, this estimate becomes 1A11

where

C is a uniform constant. Recalling that e =

obtain IA1I

where CQ =


— Z21 and

> 0, we

Bishop's equation

219

In a similar manner, we have

J

1A31

-

IC

- z2IhIdCI)

- Z21

IIuIIQ IZI — 221

C0IIulI0lzi — where For

C0 = (1 +

—

A2, we use Cauchy's theorem to deform the contour of integration to

obtain

u(

J

(for z1,z2€D).

(E8B2,—D

Since

(—

>

for

(E

ôB2f, we have 0

1

&

=

11u1101zi — Z210.

By summing the three estimates for A1, A2, A3, we obtain

IK(u)(zi) —K(u)(z2)l —

for

C0flufl0

z1,z2 ED

I

C0 is a constant depending only on This estimate is uniform for D. Therefore, this estimate holds for 21, z2 D. Together with the estimate given above for IlKu the proof of the lemma is complete. I where z1, 22

For functions u: S' —' Rd and W: S' —+ Cn_d Rd by

H(u,

=

for 0

Suppose his of class C' and letO H:

we define H(u, W): S'

<

1.

4

Ii is an easy exercise to show that

x C0(Sl,Cn_(i)

is a continuous (nonlinear) map. If h is of class C2, then H is a C' map in the sense of Banach spaces. Furthermore, we have

W)(v) =

W) . v.

The Analytic Disc Technique

220

is the Banach space derivative with respect to u which is defined

Here,

for u,v E

by

W)(v) =

H(u + tv, W) — H(u, W) t

where the limit on the right is taken with respect to the topology on the space Ca(S1, Rd). Note there is a slight loss of differentiability (i.e., if h is C2 then H is only C'). This is because the norm on Rd) involves the estimate of the fractional difference quotient. Ca(SI,IRd) is a continuous linear In view of Lemma 2, T: map. Therefore, T is also differentiable (in fact C°°) and T is its own derivative, i.e.,

(DT)(u)(v) = T(v) for u,v E Bishop's equation can be rewritten

u + T(H(u, W)) — x =0. and the vector x E W', we wish to find Given the analytic disc W: Rd the solution u: S' to Bishop's equation. THEOREM 1

SOLUTION TO BISHOP'S EQUATION

Rd is of class C2 with h(0) = Fix 0 < < 1. Suppose h: Rd x 0, Dh(0) = 0. There is a 5 > 0 such that if lxi <5 and W E C"(S', Cn_d) < 5, then there is a unique element u = u( W, x) C° (S', W) with 11W that solves Bishop's equation with u(W = 0, x = 0) = 0. In addition, if h is of x Rd —+ class C' (k 2), then there is a S > 0 such that U: C'2 (S', C'2(S1,Rd) depends in a fashion on x E Rd and W E with lxi <6 and <5. PROOF The most efficient proof of this theorem involves the Banach space version of the implicit function theorem. This is just like the usual implicit function theorem except that the domain and range are Banach spaces and norms are used instead of absolute values. We observe that the map

F: C'2(S',Rd) x C'2(Sl,CTh_d) x

C'2(Sl,Rd)

F(u, W, x) = u + T(H(u, W)) — x x is of class C' in a neighborhood of the origin in C'2(S',W') x Rd. Since Dh(0, 0) = 0, the Banach space derivative of F with respect to u

evaluated at u = 0, W = 0, x = 0 is the identity map from C'2(S', R") to itself. Since F(0, 0,0) = 0, the existence of tz(W, x) follows from the implicit function theorem. Since F is of class C', the solution u depends in a C' fashion on W and x in a If h is of class Cc, k 2, then neighborhood of the origin in C'2(S',Rd)

The proof of the analytic disc theorem for the general case

221

F is of class Ck_I and an induction argument shows that u depends in a Ck_1 fashion on x and W in a neighborhood of the origin (since (ÔF/Ou) (0,0,0) = I, differentiating the equation F(tt, W, x) = 0 allows us to solve for a jth derivative of u in terms of lower order derivatives of u). If the reader is queasy with the Banach space implicit function theorem, then the reader can use the following outline to fashion his or her own proof using the contraction mapping principle (which is the key tool in the proof of the implicit function theorem). Fix x E Rd and W E near the origin. Since Dh is small near the origin, the map

F(x,W): F(x, W)(u) = x is —

—

T(H(u, W))

a contraction on a neighborhood of the origin in Rd) (i.e., IJF(x, W)(ui) F(x,W)(u2)IICQ AIIUi — u2IIcQ for some fixed A with 0 < A < 1).

The contraction mapping principle implies that F(x, W) has a fixed point u = u(x, W), which in turn is the solution of Bishop's equation. Now, F is unifonnly Continuous in its dependence on the parameters x E and WE Since F(x, W) is a contraction, it also depends continuously on the parameters x E and W E (near the origin). In the next section, the analytic disc W depends smoothly on various real parameters. Since it depends continuously on x and W, u also depends continuously on these real parameters. We need to know that if h is of class then it depends on these real parameters in a Ck_1 fashion. This can be established (without the use of Banach space derivatives) by differentiating the equation u = F(x, W)(u) with respect to these real parameters together with an induction argument. Details of this approach are left to the reader.

15.5

I

The proof of the analytic disc theorem for the general case

We first arrange coordinates as in Theorem 2 in Section 7.2 (with k = 2) so that the given point p is the origin and

M={(z=x-4-iy,w)E Cd where h: Rd x

...,

y=h(x,w)}

is smooth (class C4) with h(0) =

0,

Dh(0) =

0

and

02h(0) (9xQOwM

As before,

= 0 for

+

1/31

= 2.

is identified with {(0,0,w);w E

and No(M) is

The Analydc DLic Technique

222

identified with {(0, y, 0); y E Rd)}. Define the bilinear form L: Cd by W3Wk, k

j,k=I

From Corollary 1 in Section 10.2, the extrinsic Levi form w

4(w)

can be identified with the map w '—+ £(w, t1) E Rd, for w E By definition, r0 is the convex hull of the image of this map. Suppose r < r0 is the cone given in the statement of the analytic disc theorem. As in Section 15.3, we can find vectors X1,. . , which lie in the image 4 with .

r < rXl...XN c is the convex cone generated by X1, .

Where

Since each X, lies in the image of 4, X3 =

a3)

.

. ,

XN.

there are vectors 1

with

j N.

For t E RN and w E Cnd, define the analytic disc

W depends smoothly on the parameters t and w Cn_d and we have W(0, = 0. From Theorem 1 in the previous section, there is a solution

u(W(t, w), x) to Bishop's equation. For simplicity, we write u(t, x, w) for u(W(t,w),x). We let A(t,x,w): D —* C" be the resulting analytic disc (see Lemma 1 in the previous section). Now h is assumed to be of class C4. From the regularity part of the solution to Bishop's equation, we see that u (and hence A) is a map of class C3 from a neighborhood of the origin in RN x Rd x to

(here, a is any fixed number with 0 < a < 1). The uniqueness part of the solution to Bishop's equation implies that u(t = 0, x = 0, w = 0)(.) = 0 and hence A(t = 0, x = 0, w = 0)(.) = 0. By suitably restricting the parameters

t, x, w, we can ensure that the boundary of the analytic disc A(t, x, w)(.) is contained in the given open set w of the analytic disc theorem. From Lemma 1 in the previous section, we have Re G(t, x, w) (( = 0) = x where G is the analytic disc with u = Re G. We summarize this discussion in the following lemma. Let O°(D, C") be the set of all analytic discs with values in C" whose boundaries are elements of LEMMA 1

Given w an open neighborhood of the origin in M, there is a 5 > 0 and a map

A: {(t,x,w) E R" x

x

ti, xl, wi <ö}

The proof of the analytic disc theorem for the general case

223

of class C3 such that the boundary of each A(t, x, w) is contained in w C M

for lxi, wi < ö. Furthermore, A(t,x,w)(() = and Re G(i,x,w)(( = 0) = x. As in the proof for quadric submanifolds, we wish to show that the set of centers

{A(t, x, w)(( = 0);

ix!, iwi <ö}

contains a set of the form for some €f >0. To do this, we + examine a Taylor expansion of A(t, x, w)(( = 0) in t for x and w fixed with In the following lemma, we identify a point (z,w) e Cd x xi, iwi x Cn_d, where z = x + iy. with (x, y, w) Rd x LEMMA 2

Given

> 0 there are constants t5 > 0,e' > 0 such that for xl, wi < 5 and

iti < €1

A(t,x,w)(( = 0) = (x,h(x,w),w) + + RN x Rd x Cn_d

where

Rd

is of class C3 and

ie(t,x, w)I S iiiti2. The sum of the first two terms on the right side of the expansion of A(t, x, w)(( = 0) is exactly the expression for the center of the analytic disc in the quadric

case (compare with (2) 15.3). This is not surprising since M can be approximated to third order at the origin by a quadric submanifold. However, the error term in the above lemma is not just any third-order error term. It is crucial for what follows that e vanish to second order in t with coefficients that are small in (x, w, t) (for example, we cannot allow a term such as to be part oLe). PROOF

Write G(t,x,w)(() = u(t,x,w)(() +

W(t,w)(( = 0) =

w,

u(t,x,w)(( = 0) =

Therefore

A(t,x,w)(( =

0)

= (x,v(t,x,w)(( = 0),w).

From Lemma 1 x.

The Analytic Dtsc Technique

224

It suffices to examine a Taylor expansion of v(t,x,w)(( = v(t,x,w)(() is harmonic in (, the mean value theorem yields

=0)

v(t, x,

0)

in t. Since

x,

Since the boundary of A is contained in M, we have h(u(t, x,

W(t,

v(t,x,w)(( =

0)

=

Therefore

=

(I)

x Note that W(t = 0,w)(() = w (a constant) and so u(t = is the unique (constant) solution to Bishop's equation in this case. Therefore,

we have

v(t = O,x, w)(( = 0) =

J h(x,w)dØ = h(x,w).

(2)

This is the constant term (in t) in the expansion of v(t,x,w)(( = 0). For the linear term, we differentiate (1) with respect to t; evaluate this at

t=0 and use the fact that u=x and W =w at t = 0. We obtain = 0,x,w)(( = 0) =

J2 Re To

save space, we have written

dØ.

for

and

Now (Ou/Ot,)(t = 0,x,w)(() is harmonic in (for (I 1 (since u = Re C). Furthermore, u(t,x,w)(( = 0) = x and so (Ou/ät3)(t,x,w)(( = 0) = 0. Therefore, the first integral on the right for

vanishes by the mean value theorem for harmonic functions. The same argument shows that the second integral on the right vanishes since w)(( = 0) = 0. Therefore, we have (3)

The proof of the analytic disc theorem for the general case

The second-order part of the Taylor expansion of v(t. x, obtained by differentiating the right side of (1). We have

jk

V

225

=

0)

in t is

(t=0,x,w)((=0)

h

= t.itk

+ e3k(t, x, w)

x, w) is a quadratic term in t whose coefficients involve the secondorder pure terms from the expansion of h, i.e., w), w)}, and (x, w)}. These pure terms vanish at x = 0, w = 0 by our choice of local coordinates (from Theorem 2 in Section 7.2). Therefore, e3k(t, x, w) can be absorbed into the error term, x. w), with the estimate stated in the conclusion of the lemma. The term (02h/OWa&G)3)(X,W) can be written where

02h(0,0)

+

(82h(x,w)

U2h(0,0)

The term in parentheses can be made as small as desired by suitably restricting x and w. Therefore

(t = 0,x,w)(ç = 0) = + where where

is the quadratic form that generates the Levi form at the origin and

provided (t, x, w) belongs to a suitably small neighborhood of the origin.

We have

ow

=

Substituting this into £ and integrating ç =

over

0 0 for

27r, we obtain

226

The Analytic Disc Technique

When j = k, we have (for 1(1 = 1)

=

=x2. Therefore, from (4), we have

t,tk

82V

—o

2

—

f w)

—

if j = k.

Since the third-order Taylor remainder in t can be absorbed into the error term, the proof of the lemma follows from (2), (3), and (5). I Let us summarize where we stand. We have shown (Lemma 1) that each

point in the set

Il = {A(x, w, t)(ç = 0); iti, lxi, wi belongs to the image of an analytic disc whose boundary is contained in the given

open set w for the analytic disc theorem. Furthermore, the Taylor expansion of t in Lemma 2. The constant term in this expansion A(x, w, is (x, h(x, w), w) and the set wr = {(x, h(x, w), w); lxi, wi < 6} is an open subset of M that contains the origin. Therefore, the proof of the analytic disc theorem will be complete once we show that for fixed x, w with xl, wi < the map t

f(x. w)(t)

tE

= parameterizes a set that contains an open neighborhood of the origin in the cone

ti < es', r (here, €' and are as in Lemma 2). Now the map t that parameterizes an open neighborhood of the origin of the cone XN. The hope is that since contains the cone I' by the choice of f(x, w)(t) will x, w)i is small relative to ti2 (Lemma 2), the image oft also contain the desired neighborhood of F.

To carry out the details, we replace t, by

for t, 0. We define two

maps

and

F(x, w)(tj,

.

. . ,

tN) = f(x,

. .

.

,

E(ti,...,tpj) +e(t,x,w) where the error term e(t, x. w) =

x, w) is continuous (but not

The proof of the analytic disc theorem for the general case

227

differentiable at t = 0) and satisfies the estimate

ie(t,x,w)i provided lxi, iwl <6 and iti <((?)2/N). Here, > 0 can be chosen as small as desired and 6 depends only on We wish to show that 6 can be chosen so that if lxi, iwi <6, then the image of the set {t E RN; t3 0 ti < ((€')2/N)} contains

an c-neighborhood of the origin of r. Recall that r < rXI...XN. Therefore, for each v E r n S (S is the unit sphere in Rd), there is a conical neighborhood of v, denoted . . . , and a collection of d-linearly independent vectors with E

Since ms is a compact subset of rXI..XN, we can cover r by a finite number of such Therefore, it suffices to show the following lemma. LEMMA 3

Sup,ose X1,... , are linearly independent vectors in Rd. Suppose F(t) = t3X3. Suppose r < rXl...Xd and c' > 0 are given. Then there exist Rd is a continuous map with iF(t)—E(t)i '1 it! 17, e > 0 such that F : Rd for t = (t1,. .. , td) with t3 0, 1 j < d, and < €', then the image of

{t=(tl,...,td);t3Oiti<e'}underFcontainsBEflr.

PROOF

In our context, F is differentiable except at t = 0. We could use

the inverse function theorem to examine the images of little balls which are contained in the set rx, Xa• However, the proof of the above lemma does not require any differentiability assumptions on F. So we offer this purely topological proof. We may assume that the vectors X1,. . . , Xd are the standard basis vectors in Rd. Therefore, we have rXI...Xd = {(x1,. . . , xd); x3 0). Given a cone such that if > 0 and there exists 0 < < rXl...Xd there exists x e r n BE then the Euclidean distance from x to O{rXl...Xd fl B4 is greater than ii'ixi. Since X1,. . . , Xd are the standard basis vectors, F is just the identity map. e > 0 are chosen small relative to and Suppose it — F(t)i S iiiti. If €', then the line segment between t E O{rXl,..Xd fl and F(t) does not intersect (see Figure 15.2). Now suppose the point XE rflBE is not in the image of FXI...Xd fl under F. By the above discussion, the restrictions of F and the identity map to O{rX,...Xd fl are homotopic in Rd — {x}. Since fl is homeomorphic to a (d — 1)-dimensional sphere in Rd which encloses x, the homology class of the image of O{rXI.,.Xd fl } under F is nontrivial in the (d — 1)st dimensional homology group of Rd — {x). On the other hand, rXI...Xd fl BE' is contractible to the origin in Rd. Since F

r

228

The Analytic Disc Technique

t2

image of fl

under F t

€

FIGURE 15.2

is continuous and x is not in the image of F, the image of O{FxI...xd fl under F is also contractible in — {x}. This means that the homology class of F {O{['x..xd fl is trivial in the (d— l)st dimensional homology group of — {x}. This contradiction proves the lemma. I stated earlier. Lemmas 2 and 3 complete the proof of the analytic disc theorem, which in turn completes the proof of the CR extension theorem. As

16 The Fourier Transform Technique

In this chapter we present a Fourier transform approach to the proof of the CR

extension theorem. This technique has the advantage in that it can be more easily adapted to the holomorphic extension of CR distributions, which will be discussed at the end of Part ifi. However, the goal of this chapter is to introduce the technique rather than to prove the most general theorem. Therefore, to avoid some of the cumbersome technicalities of more general results, we shall assume where d = codimR M the given CR function is sufficiently smooth (class will suffice). In addition, we shall assume that the submanifold M is rigid,

which means that near a given point p E M there is a local biholomorphic change of coordinates so that p is the origin and

M = {(z = x+iy,w) E Cd

x Cfl_d;

= h(w)}

with h(O) = 0 and Dh(0) = 0. where h: Cn_d is smooth (say class The point is that the graphing function, h. for a rigid submanifold is independent of the variable x E Rd. The modifications required to handle the more general case where h depends on both x and w will be mentioned at the end of Part ifi. The basic idea of this technique is to use the approximation theorem given in Chapter 13 to derive a modified Fourier inversion formula for smooth functions on M. We will then show that this modified inverse Fourier transform of a given CR function is the restriction of a holomorphic function provided the modified Fourier transform of the CR function is exponentially decreasing. Finally, we show that the set of directions in which the modified Fourier transform of a CR function is exponentially decreasing is related to the convex hull of the image of the Levi form and the CR extension theorem will follow. The ideas presented in this chapter are due to Baouendi, Treves, Rothschild, Sjostrand, et al. (see [BCT], [BRT], [BR2], [Sj]). Our presentation is closest to that in [BRT].

229

The Fourier Transform Technique

230

16.1

A Fourier inversion formula

Our desired Fourier inversion formula will be derived after we present three lemmas. The first of these lemmas is analogous to Lemma 1 in Chapter 13 for the approximation theorem. Instead of an integral over totally real n-dimensional

slices of M as in Chapter 13, we integrate over the following d-dimensional slices. For p = (z, w) E Al let

=

M,,

= {(z',w) E JVI;z' E

Our analysis will be a local one about the origin. So as in Chapter 13, we assume the graphing function h: Cn_d ...+ is suitably cutoff so that < 1/2 on LEMMA I

and Suppose U1 and Uj' are neighborhoods of the origin in C" with U1' Cc Let g be an element of suppose U2 is a neighborhood of the origin in 1 on Uj'. Suppose f: {U1 x U2} fl M —* C is a continuous V(U1) with g function. Then for (z, w) E {U1' x U2} fl M,

f(z, w) = urn

f

g(z')f(z',

z' E

+ and [z — z'12 = (z1 — Moreover, this limit is uniform in (z, w) E {Uj x U2} fl i'vf. where

dz' =

A ...

A

...

+

—

Note the function f is not assumed to be a CR function. We will not assume f is CR until the next section. The purpose of the cutoff function g is to ensure that the integrand has compact support. The reason we do not assume f has

compact support is that we will apply this result in the next section to CR functions, which typically do not have compact support.

PROOF The proof is the same as the proof of Lemma 1 in Chapter 13. The only difference is the dimension of the slice of M over which we are integrating. This changes the constants involved but otherwise has no effect on the proof. Also note that c here plays the role of in Chapter 13. I LEMMA 2 Suppose U1,

U2 andg,f areas in Lemma!. Thenfor(z,w) E

f f where

the limit is uniform for (z, w) E {U( x U2} fl A'!.

A Fourier inversion formula

w=

Here, w u = PROOF

231

= (ul,...,nd) E

(w1

Cd.

by 4f in the statement of Lemma 1. We obtain

Let us replace

f(z,w) =

J

z' E

Lemma 2 will follow once we have established the following identity:

= (f)d/2

f

(1)

To show (1), first note that if w is a vector in Rd (rather than C'), then by a translation, we have

J

=

f

(2)

The left side of this equation is an entire function of w E C' which agrees with the right side when w E Rd. Therefore, equation (2) holds for all w E C' by the identity theorem for holomorphic functions. Now we complete the square in the exponent to obtain

=

J (by letting w =

—

z') in (2))

= The last equality follows from a standard polar coordinate calculation. Multiplying this equation by yields (1) and so the proof of the

lemma is complete. For a continuous E 1W', define

I

function f: M

=

J z' E

C

and for z E Cd, w E

dz'.

and

The Fourier Transform Technique

232

Lemma 2 can now be rewritten

f(z,w) =

urn J

We will show in Lemma 3 below that if f is of class +

on M,

for some uniform constant C. Since E Rd. the dominated convergence theorem

+ is integrable in

then implies

f(z, w)

=

f If(z, w,

for (z, w)

x U2} n M.

eERd

This is analogous to the standard Fourier inversion formula for Euclidean space.

The only difference is that If involves an integral over a slice of M (rather than Euclidean space). Also, it is customary in Euclidean space not to include in the integrand of the Fourier transform as we have done in If. the factor would then reappear in the Fourier inversion formula. The factor For technical reasons which will become clear, we wish to modify the above — z']2 in the exponent. Fourier transform to one with a term of the form Rd define For z E Cd, wE Cn_d, and

If(z, w,

f g(z')f(z',

— z',

z' E

= (1 + i(u .

where

for

E Cd.

LEMMA 3 There exist neighborhoods

and Uj' of the origin in Cd with U CC Uj and a neighborhood (12 of the origin in Cm_d such that if g D(U1) with g = 1 on U, then the following holds. (a)

1ff: Mn {Ui x U2} —'

C

is afunction of class Ca', then there is a

constant C > 0 such that + I)_(d+1) for all

I(If)(z,w,e)I

+ 1)_(d+I)

forall

E Cd

E Rd

and for

(z,w) = (x+ih(w),w) E Mn {Uj' x U2}. (b) More generally, if f: Mfl{U1 xU2} —+ C isafunction of class CN,N> is a jth order derivative (0 j N) in on Ctm, 0, and if then there is a constant C > 0 such that +

for alle E Rd andw E U2,z —x+i(h(w)+v) E Uj' (here v

R").

A Fourier inversion formula

233

We will need the above estimate in part (a) on lf(z, 1L', for E Cd (rather than just e E Rd) because in the proof of the next theorem, we will transform the k-integral over IRd in the Fourier inversion formula for If into an integral over a contour in C" which will yield a Fourier inversion formula for If. In the next section, we will need the more general result in (b), which estimates the derivatives of If for (z, w) in a -neighborhood of the origin.

The proof of the estimate for If in part (a) is a special case of part (b) d + 1, and v = 0). Therefore, we first prove the estimate 0, N l}. So it in (b). This estimate clearly holds on the compact set E suffices to show PROOF

(with j =

1D3{(If)(z,w,e)}I

for

1.

For fixed w E Cn_d the set is parameterized by the map z' = x'+ih(w) for x' For z = x + i(h(w) + v) with v E Rd. we have

(If)(z,w,e) =

f

— x' +

x' ERd

= g(x' + ih(w)) and J(x',w) = f(x' + ih(w), w). If jth order derivative in z, w, ü), then where

{(If)(z, w,

=

f

is a

z,

(3)

Gk (f) is an expression involving kth order derivatives of Moreover, Gk(f)(x', z, has compact x'-support and is a polynomial in and where

of degree j — k (as a result of the derivatives of order j — k of the exponential term).

Now the idea is to integrate by parts with the vector field

As we will see, each integration by parts will yield a factor of this procedure N — k times will yield the lemma. To carry out the details, note

L{ie. (x — x' + iv) — IeJ[x — x' + iv]2} = where

+

x',

Iterating

The Fourier Transform Technique

234

The term

is

homogeneous of degree one in

and satisfies the estimate

+ Ix'I + for some uniform constant C > 0. Choose neighborhoods U1 C Cd and U2 c

of the origin so that if w E U2 and if z = x + i(h(w) + v) and z' =

Cn—d

x' + ih(w) belong to Ui, then

We have (4)

U cc U1

Let

g = 1 on

be a neighborhood of the origin in Cd. Let g e D(U1) with

We have L1

(x—x +zv)—feI[x—z +ivj 2

=

+ 7j(x, x',

(5)

Substituting the left side for the right and integrating by parts, we obtain

f

z,

x' ERd

=

—

I

J

LI

Gk(f)(x', z,

1.

I iei

J

Using (4) together with the fact that Gk(f)(x', z, and ofdegreej — k, we have

Gk(f)(x',z,w)(e) <

is a polynomial in

for

>

1

—

—

C is some uniform constant. Since L is a differential operator in x' whose coefficients are homogeneous of degree zero in we have where

s

for

1.

is homogeneous of degree j — k in whereas that IGk(f)(x', z, If f is of class CN, the above term is homogeneous of degree j — k — in then Gk(f) is of class and we may iterate the above procedure starting

Recall

1

A Fourier inversion fonnula

235

with (5) N — k times to obtain

J G,,(f)(x', XIERd

=

f

z,

(6)

x' ER"

is homogeneous of degree j — N in

where

for where C is a constant independent of w

Therefore

1

U2 and z = x + i(h(w) + v)

(7)

U.

For the exponential tenn, we have I=

I

eRe{1 (xx'-4-:v)

= < This estimate together with (6), (7), and (3) yield part (b) of the lemma. As already mentioned, the estimate on If given in part (a) is a special case of part (b). The estimate on If in part (a) is proved in a similar manner. In fact, this estimate is easier since the exponent occurring in If is simpler. Therefore, it will be left to the reader. I We

remark that the estimate in part (b) also holds for e in a

neighborhood of Rd. We now state and prove the Fourier inversion formula for the transform If. THEOREM 1

There exist neighborhoods U; cc U1 of the origin in and a neighborhood 1 on u; and ('2 of the origin in such that if 9 E 'D(Ui) with g

f:

M —* C is of class Cd+ 1, then

f(z,w)

E

{U x U2}nM.

=

J

The Fourier Transform Technique

236

PROOF From Lemma 2 and Lemma 3 part (a) and the dominated convergence theorem, we have

f(z,w)

f

=

R

=

urn

R—.oo

R

f f (?f)(z, w,

-R

..

(8)

.

—R

For (z, w) e M near the origin, we write z = x + ih(w) for some x E Rd. The slice is parameterized by z' = x' + ih(w), for x' e Rd. Therefore

=

J

x' ERd

where as before w) = g(x' + ih(w)) and f(x', w) = f(x' + ih(w), w). Note that is an entire function of e E C" (since g has compact x'-support). We can use Cauchy's theorem to change each in (8) to an integral over the contour — e R} in the complex = + plane (1 = R} appearing j d). The integral over the side contours in the change of contour process disappear as R —' 00, because the measure whereas the integrand is O(R_(d÷l)) by the of these side contours is

estimate on If in Lemma 3. By replacing I

by

+

—

x') in the

exponential term in If, we obtain

=

(9)

which is the exponential term in the definition of If. Moreover, we have

d

iC(x—x') '

=

—

x',e)de.

)deiAdei+IA...Aded

('°)

237

The hypoanoiytic wave front set

From (8) and Cauchy's theorem, we obtain

f(z, w) = lim J 7 If(z, w, =

iimf

7 I

R-.ocJI

= urn

=

(with

=+

- x'))

(by (9) and (10))

J

J

This completes the proof of Theorem 1.

I

The hypoanalytic wave front set

16.2

To summarize our progress so far, we have shown in the last section (Theorem 1) that

f(z,w)= f

for

(z,w)EMneartheorigin

(1)

where

(If)(z, w,

1ff:

=

Al —+C is of class + l)_(d+1) for

f g(z')f(z',

— z',

then we have shown (Lemma 3) that I(If)(z, w, E

and (z,w) €M and hence the integral in (1) is

well defined. Note from its definition that (If)(z, w, is analytic in z ECd. If (If)(z, w, is exponentially decreasing in E Rd, then the right side of (1) also defines an

analytic function of z E Cd. Later, we will see that if f is a CR function near the origin on M, then the right side of (1) also defines an analytic function of w E Cn_d near the origin. In this case, (1) shows that f is the restriction of an ambiently defined holomorphic function. All of this is to serve as motivation for examining the set of vectors e E Rd in which (If)(z, w. is exponentially decreasing. Roughly speaking this is the complement of the hypoanalytic wave

238

The Fourier Transform Technique

front set of the function f at the origin. More precisely, we make the following definition. Fix a smooth function g: Cd —p

JR

with compact support which is identically

one on a neighborhood of the origin.

Let F be a set of continuous functions on M. A vector is not in the hypoanalytic wave front set of F at the origin if there exist DEFINITION 1

E Rd

a cone F in JRd containing (b) a neighborhood U of the origin in (c) a constant 0 (a)

such that if f belongs to F, then there is a constant C > 0 such that <

for e

er,

(z,w) E U.

In our application, F will be the class of CR functions of class Cd+2 on an open neighborhood w of the origin in M. However, the definition does not require the function f to be CR, which is the reason for the more general definition.

The order of the quantifiers is important. The cone 1', the neighborhood U in and the constant e > 0 are independent of the function f E F. The constant

C > 0 is allowed to depend on f. In the literature, it is more common to see the concept of the hypoanalytic wave front set of a single function. However, we wish to holomorphically extend all CR functions defined on an open set w to a fixed open set in C'. For this reason, we have modified the more standard definition to the one given above. We leave it as an exercise to show that the above definition is independent of the cutoff function g. This follows easily from examining the term — z')2

in the exponent of the definition of (If)(z, w, We denote the hypoanalytic wave front set of F at the origin by WF0(F). The set is closed in Rd. We identify this copy of JRd with the space of vectors that are normal toM at 0, denoted N0(M) = {(O,y.O);y E JRd}. In the literature, WF0(F) is often considered part of the totally real tangent (or cotangent) space of M at 0, (X0(M)), which in our coordinates is the space { (x, 0,0); x E

JRd

However, since we are extending CR functions in directions

that are normal to M, it will be more convenient for us to think of WF0(F) as a subset of No(M) rather than Xo(M). It is possible to define WF,(F) for any point p E M, by first using a coordinate change so that p 0 and so that M is graphed above its tangent (F) is also available. space as we have done. A more invariant definition of However, we shall not use this definition. Instead, we refer the reader to [BRT] or (BR2J.

For an open set w C M containing the origin, let CR(w) be the set of CR is on w. If WF0(CR(w)) is empty then

functions of class

The hypoanalytic wave front set

239

exponentially decreasing for all E Rd and all I e CR(w). In this case, we will show that each f E CR(w) can be holomorphically extended to a neighborhood of the origin. We also wish to show that if WF0(CR(w)) is contained in some cone, then elements in CR(w) can be holomorphically extended to some open subset which lies to one side of M, as in the conclusion of the CR extension theorem stated in Chapter 14. To state this criterion for holomorphic extension, we need some additional notation.

For a cone F C Rd, define the polar of F, by

forall Note that ['0 is a closed convex cone in Rd. It is also easy to show that is the closure of the convex hull of I' (this uses the fact that any point not in a convex set can be separated from it by a real hyperplane). THEOREM I

Let w C M be an open neighborhood of the origin in M. Let F C N0(M) be a closed convex cone. If the interior of is nonempry, then the following are equivalent: (a) (b)

WF0(CR(w)) is contained in F. For each open cone < fo, there is a neighborhood w1 of the origin in M and there exists > 0 such that for each f E CR(w), there is a holomorphic function F defined on the set w1 + {F, fl } which is continuous up to w1 with = f.

The proof of this theorem will show that even if f is not CR, then part (a) implies that f extends to a function that is holomorphic in z Cd (but not wE Cn_d) for (z,w) E w1 In the next section, we will relate the hypoanalytic wave front set to the convex hull of the image of the Levi form. This relationship together with the (a) (b) part of the above theorem will complete the proof of the CR extension theorem. Let W C ill be a neighborhood of the origin. Let Ui, Uj', and U2 be the open sets that satisfy the conclusions of Lemma 3 and Theorem 1 in the x U2} fl M C W. Since the definition previous section. We also require of WF0(CR(W)) is independent of the cutoff function g, we choose g e with g = 1 on U. PROOF

We first show that (a) implies (b). We start with the Fourier inversion formula

f(z,w) (z,

E

=

J

Mn {U1' x U2}. We wish to show that if WF0(CR(w))

is contained in a convex cone F C No(M), then the right side of (2) extends analytically to an open set of the form W1 + {F1 fl } where I'I is any given

240

The Fourier Transform Technique

C M and > 0 both depend on Fi. If r1 0 and an open cone F which is slightly larger than F (i.e., F
vErt,eEr.

for The

(3)

integral in (2) can be split into two integrals to obtain

f(z,w)=

E

Mn

J x U2}

Now {IRd — F) n S is a compact subset of the complement of the hypoanalytic wave front set. Using Definition I together with a finite open covering argument,

we can shrink U1 x U2 c such that

if necessary and find constants C. > 0

Cd x

for

(z,w) E U1 x

U2,

E Rd — F.

(5)

is holomorphic in z E Cd, an application of Morea's theorem

Since

and Fubini's theorem shows that F2(z, w) is holomorphic in z for (z, w) E U x U2. Next, we show that F1 (z, w) is holomorphic in z E Cd for (z, w) e + n where > 0 have yet to be chosen. F1 WI and { } Let w1 = x (J2} n M. Points in the set w1 + F1 are the form (z, u')

(x + i(h(w) + v), w)

where

(x+ih(w),w)Ewi

and

vEF1.

From part (b) of Lemma 3 from the previous section with N = d + 2 and

j=

1, we have

w,

D

is a uniform constant

that is independent of z = x + i(h(w) + v) E Uj' with V E F1 and w E U2. Using (3), this estimate becomes

ID{(If)(z, w, for

E r and V E F1.

+1

The hypoanalyac wave front set

= Let we have lvi

241

>0. For (z,w) = (x+ih(w),w)+(iv,0)
wi

+{r1

for

E F.

Due to the exponential decay in this estimate and the analyticity of (If)(z, w, in z E Cd, the function

Fi(z,w)

-

f + {F1 fl

}. The above estimate because + 1)_(d+1) E Rd. Since we have already shown that F2(z, w) is holomorphic

E

E

on ID{If}i also implies that F1 is C' up to integrable in

in z E Cd for (z, w) E Uj' x U2, (4) shows that f is the restriction on w1 of the function F(z,w) = Fi(z,w) + F2(z,w) which is holomorphic in z E Cd for (z,w) E w1 + (F, flBf1} and C' up to w1. So far, we have not used the fact that f is CR. The next lemma shows that if f is CR, then F(z, w) is also holomorphic in w Ca" for (z, w) {r1 fl } and thus the proof of (a)

(b) will be complete.

LEMMA 1

Suppose I is a CR function of class C' defined on the set w1 C M. Suppose } and that F is F(z, w) is holomorphic in z Cd for (z, w) E + {I'1 fl F is also holomorphic in w Cn_d and thus C' up to w1. If F is the holomorphic extension off on W, + {F, fl }. PROOF As shown in Theorem 3 in Section 7.2, a local basis for H°"(M) is given by

..

,

d

as well as on M. Since

These vector fields are defined on the ambient F(z,w) is holomorphic in z OF

—

(L,F)(z,w) for (z,w)

ijWj

w, + {F, flBEI}.

Furthermore, since F is C' up to WI and f is CR on

we have

Owi

=0

So for fixed w, (z,w) WI + {F, n

(since I is CR).

w) is a holomorphic function of z Cd for and (OF/OiD,)(z,w) = 0 on w1. Now WI is

The Fourier Transform Technique

242

Cn_d near the origin. There+ i(h(w) + y),w) is holomorphic in z = x + iy C" for fore, {r1 fl } and vanishes when y = 0. By the identity theorem for holomorphic functions, OF/OzD3 must vanish identically on w1 + {F1 fl } and so fl as desired. F is holomorphic on w1 + }, Note that this version of the identity theorem follows from the elementary parameterized by (x + ih(w), w) for x E Rd, w

theory of analytic functions of one complex variable. To see this, fix any vector v Fj C No(M) with VI = 1. Then Jv is a totally real tangent space w) is a holomorphic vector. The function G(x+iy) = which vanishes for y = 0. Therefore function of x + iy C with 0 < y < by the Schwarz reflection principle and G(x + iy) = 0 for all 0 y < identity theorem for analytic functions of one complex variable. Since v r1 with lvi = I is chosen arbitrarily, we have OF/thz)3 0 on W1 + {r1 n as desired.

As mentioned just before Lemma I, this lemma completes the proof of (b).

(a)

For the converse, we assume (b) holds — that is, we assume there is a closed, convex cone F C No(M) which satisfies statement (b). We must show that if WFO(CR(W)). Since F is closed and convex, we have F then {r'°}° = F. Therefore, if does not belong to F, then there is a vector belonging to the interior of P with ivol

= I and

v0

<0.

(6)

Choose an open cone F1

/

CR(W) extends to a holomorphic function F defined on the set

+

fl

}C

> 0 and where w1 is some open neighborhood of the origin in M. We where may assume Wi is of the form

= {(x+ih(w),w);x

IR'1,w

with

lxi,lwi < 6}

for some 5 > 0. Now the idea is to deform the domain of integration in If using Cauchy's theorem and then estimate the resulting integrand. To carry out the details, we 36/4 and choose g V(Cd) such that if ivl
then g1(x + iy) = 0 < t <e1/2, let

I

for lxi S 6/2 and g1(x + iy) =

= {z' = x' + i(h(w) + where

+ {r1 fl ix'i 36/4).

0

for lxi 36/4. For x'

R"}

C = g1(x' + ih(w)). Note that if ih(w)i q/2, then (since gi(x',w) = 0 if fl {ix'J 36/4} C and

The hypoanalytic wave front set

243

= f, we have

Since

(If)(z, w,

=

J F(z', w)g(zl)e

— z',

z'

For fixed w with ih(w)I

the integrand is holomorphic in z' for z' = 35/4, lvi ei/4 and v r1 (note g(z') = here). Therefore by Cauchy's theorem, we can deform the domain of integration from to to obtain

x' + i(h(w) + v) provided lx'i

(If)(z, w,

=

f F(z',

— z',

z' E Mt.

provided Let

ih(w)i

fi/4

and

0 < t < Ei/4. =

(z — z')

—

—

zh}2

which is the exponent appearing in

the integrand. The variables z E C1 and w E are independent variables. The variable z' depends on w, the independent variable x', and the parameter t through the equation z' =

x' + i(h(w) + (x', w)v0) E To show that the vector eo does not belong to WF0(CR(w)), we must choose t and show that there is an €> 0, a conical neighborhood r of in and a neighborhood U of the origin in such that

Re{q(z, z', w)(e)}

(7)

for F, (z.w) E U, and z' fl {iRe z'I <5}. Note that since q does not involve the function the choice of €, F, and U will be independent of 1. It suffices to show that there exist > 0 and 0 < t < /4 such that

Re{q(z = O,z',w =

(8)

{Mt,w0}fl{iRe z'I <5}. This is because q is homogeneous of degree and continuous in (z, w) and x' and therefore the inequality (7) will hold with replaced by for (z, w) in a neighborhood of the origin in and for in a conical neighborhood of A point z' E fl {iRe z'I <ö} is of the form for z' 1 in

= x' +

with

ix'I <

We obtain

Re{q(z =

0, z',

w=

= Im{z'} =

—

Re{[z']2}

(x', 0)vo

—

(9)

The Fourier Transform Technique

244

From (6), recall that v0

< 0 and vol = 1. If

6/2, then

z', w =

+

=

I

and (9) becomes

Re{q(z =

If 6/2 < x'I

0,

t(v0

(lOa)

6, then (9) becomes Re{q(z

(lOb)

<0, we can choose t suitably small with 0 < t < so that the vO right sides of (lOa) and (lOb) are both negative. l'his establishes (8) and by the discussion after (8), does not belong to WF0(CR(w)), as desired. The proof (a) is now complete. I of (b) Since

The hypoanalytic wave front set and the Levi form

16.3

In this section, we relate the convex hull of the image of the Levi form and WF0(CR(w)), thereby completing the Fourier integral approach to the proof of the CR extension theorem. Let F0 be the convex hull of the image of the Levi form of M at 0. Let F = {r0}0 (i.e., the polar of the closure of F0). Note that P = r0. According to Theorem 1 in the previous section, the CR extension is contained in F. This theorem will follow once we show that is the content of the next theorem. THEOREM I Suppose F is the polar of the closure of the convex hull of the image of the Levi form of M = {y = h(w)} at the origin (i.e., F = {Fo}°). Then,for any neighborhood w C M of the origin, WF0 (CR(w)) is contained in r.

Given F, we must show there is a neighborhood U of the origin a conical neighborhood F of in R", and a number 0 such that if f is an element of CR(w) then there is a constant C > 0 such that

PROOF

in

I(If)(z,w,e)l for all (z,w) EU and E F. with {(O,w);w Cn_d} and No(M) with be the extrinsic Levi form of Mat 0.

As usual, we identify {(O,y,O);y E W'}. Let We have

4(w) =

Wj'tt'k

j,k=I

k

for

w = (w1,. .

.

,wnd) E

The hypoanalytic wave front set and the Levi form

By definition, the cone F0 is the convex Since F, is closed and F° =

cone,

245

(in Rd) of the image of The I', there must exist a vector v0 F0

hull

with

v that lie in the image of then the same inequality holds for all v in the convex hull of the image of Therefore, we may assume (1) holds for some vector v0 that lies in the image of By a complex linear change of coordinates in the w-variables, we may assume v

V0

= A(ei)

where

=

(1,0,... ,0) E Ca"

Now we examine the second-order Taylor expansion of h about the origin. We may assume there are no second-order pure terms in this expansion (Theorem 2

in Section 7.2). Therefore

h(w) = £o(w) + 0(3) where 0(3) involves terms that are third order in w and th. Let V)1 = (w2,.. ,Wn..d) e We have

=

h(w)

= Since

a>

0

<0, there is

v0

+ 0(3)

+ wij2vo

+

0(Iw'IlwE)

+

0(3).

a conical neighborhood F of

in

Rd and a number

such that

for

We obtain

h(w) .

<

+ C}w'IIwI + Ciw'13)

for

EF

where C is a positive constant that is independent of E I' and w E the origin. Now we parameterize by = x'+ih(w) for x' E Rd. With z Cd, we obtain

= (27r)_d

f —

x' + i(y — h(w)), e)dx'

near =

x+iy E

The Fourier Transform Technique

246

where

w) = g(x' + ih(w)) and J(x', w) = f(x' + ih(w), w). Let

q(x,x',y, The estimate on Re

=

(x —

x' + i(y — h(w)))

— x'

—

+ i(y

—

h(w))]2.

requires an estimate on Re q. We have

= h(w)

—y

—

—

xtI2

+

—

h(w)12

+CIw'I(wl +CIw'13

+

+ 2(1y12 + Ih(w)l2))

the last inequality follows from (2). We restrict (z, w) E so that <&IwiI = rand w'I
Re q(x, x', y,

+ C6r2 + C83r3 + S +

+ 4C2r4)

First choose r > 0 small enough so that (—a/2)r2 + 4C2r4 <0. Then choose

S > 0 small (depending on r) so that

+ 4C2r4 + [C6r2 + C63r3 + S + Denote the number on the left by —f (with c > 0). We obtain for It follows that (3)

fore EF and (z,w) E with IzI <6, Iwil = r, lw'I <Sr. To prove that does not belong to WF0(CR)(w)), we need to establish (3) for Iwi I
There is a neighborhood U of the origin in C" and a constant e > 0 such that C which is analytic

for each f E CR(w) there is a function G(f): U x Rd

The hypoanalytic wave front set and the Levi form

247

in (z, w) E U and a constant C > 0 such that

I(If)(z,w,C)

for all (z,w) EU and

—

E Rd.

Here again, the open set U and the constant are independent of f E CR(w). However, the constant C is allowed to depend on f. Assuming the lemma for the moment, we complete the proof of Theorem 1 and hence the proof of the CR extension theorem. The estimate in (3) together with Lemma 1 implies that there is an e > 0 so that for each f E CR(w) there is a constant C > 0 such that

IG(f)(z,w,C)I with IzI <6, wit = r,andtw't <örprovidedr >0 fore E rand (z,w) and 6 > 0 are chosen suitably small as above. Since G(f)(z, w, C) is analytic in (z, w), the maximum principle implies that the above inequality also holds for IwiI < r, and Izi < 6, w'I < 6r. This together with another application of the lemma yields the estimate

forCE F and (z,w) E with lzI <& Iwil r, and Iw'I <6r. By definition, the vector E r does not belong to WF0(CR(w)), as desired.

Since(If)(z,w,C) is already analytic in z E Cd, the idea is to solve the appropriate a-problem in the w-variables to find the appropriate analytic G(f)(z, w, C). From the formula for 1(f), we may write PROOF OF LEMMA 1

(If)(z,w,C)

=

x'

f

E(z —

—

where

E((,C) = is

a holomorphic function of ( = g(x' ± ih(w)). Differentiating If,we obtain

Cd. As usual, f(x',w) = f(x' + ih(w),w)

and

w,C)

-

f

=

J

E(z -

- -

k=I

(4)

The Fourier Transform Technique

248

The second term on the right can be rewritten as

if

k—i

After integrating by parts, this term can be combined with the first term on the

right side of (4) to yield

=

f

E(z-x' -

where

3

(M) is given by L1,..

From Theorem 3 in Section 7.2, a basis for

.

,

Ln_d

where

=

ôhk(w)

—

Ow3

Ow3

4

j

1

OZk

Let it:

betheprojectionir(z'

From the definitions of f and

note

that f = / o it

n

- d.

=x'+iy',w)

and

g=

o

(x',w).

ir. A)so note

1<_j_
1

on a neighborhood of

= ir..L3{fg} = L3{fg}

=0 near the origin in M. So there exists 6 > 0 such that if Izi, wi < 6, then the <6. integrand in (5) vanishes for Therefore, to estimate (5), we only need to estimate the real part of the exponent, q, for Ix'I

q(z, x',

6, where

=

— ih(w))

(z —

—

—

—

ih(w)]2.

With z=x+iy, we have Re{q(z, x',

= (h(w)

—

y)

—

— xh12

— It' — h(w)12).

The hypoanalytic wave front set and the Levi form

Re q(z = O,x',w =

249

<

q is homogeneous of degree one in and continuous in z, x'w, there is a neighborhood U of the origin in en such that if (z,w) EU and 6, then Since

Re Note

6 is independent of f E C.'R(w), x' E Rd with From this estimate and (5), we have

6, (z, w) e

U and

Rd. Here, C1 is a constant that depends on f but it is independent of (z,w) E U and E Rd. By shrinking U, we may assume that U = U1 x U2 where U1 is a neighborhood of the origin in Cd and U2 is a strictly convex neighborhood (such as a ball) of the origin in en d• Since for (z, w) E U and e

is Ow-closed in U2, we can use the Ov-theory on strictly convex domains to find a solution K(f)(z, w, to the equation

= with the estimate

C sup

sup wEU2

wEU2

for z E U1,

E Re'. Here, C is a uniform constant that is independent of Rd, and f E CR(w). For the reader who is not familiar with the z E U1, E solution to the 0-problem on a strictly convex domain with sup-norm estimates, an integral kernel solution to this problem is provided in Part IV (see Theorem 1 in Section 20.3). Combining (6) and (7), we obtain

for (z, w) E U1 x U2 and e Here, C depends on f but it is independent of (z, w) E U1 x U2 and E Rd. From the integral kernel formula for K(f)(z,w,e), this solution is holomorphic in z U1 (since is holomorphic in z

G(f)(z, w,

U1). Define

is analytic in both z E Ui and w E U2. From (8), we = 62/2.

obtain the desired estimate for the lemma, with

250

The Fourier Transform Technique

This completes the proof of Lemma I. As stated prior to the statement of Lemma 1, the proof of Theorem I is now complete. As stated at the beginning of this section, Theorem I together with Theorem 1 in the previous section completes the Fourier integral approach to the proof of the CR extension theorem.

— z']2 in the It is worthwhile to note the key role played by the term Without this term, one would not be exponent of the transform (If)(z, w, able to prove estimate (6) in the proof of the above theorem or estimate (lOb) in the proof of Theorem I in Section 16.2. This is the reason for using the transform (If) (z, w, rather than the simpler transform (If) (z, w,

17 Further Results

In this chapter, we discuss some extensions of the techniques developed earlier

in Part ifi. We begin by outlining the modifications needed for the Fourier transform approach to the proof of the CR extension theorem for nonrigid CR manifolds. Next, we discuss the holomorphic extension of CR distributions. The chapter ends with a discussion of CR extension near points of higher type and analytic hypoellipticity.

17.1

The Fourier integral approach in the nonrigid case

In Chapter 16, we present the Fourier integral approach to the CR extension theorem in the rigid case M {y = h(w)}. As usual, our coordinates for are given by (z, w) where z = x + iy E Cd, w E Cn_d with d = codimR(M). In this section, we outline the major changes needed for the nonrigid case M = {y = h(x,w)} and we leave some of the details to the reader (or see [BR2]).

As in Section 16.1, we start with the transform

=

dz'

J

is a continuous function on M and where g is a smooth function on 0, let M with compact support which is identically 1 near the origin. For

where /

=

where we have used the notation

f

de

=

analytically continues to

Note that the map + ... + E Cd}. Lemmas 1 and 2 from

251

Further Results

252

Section 16.1 hold without change to show

as€'—'O

for (z, w) e It'! near the origin. For each e > 0,

zECd.

is an entire function of

The first change needed for the nonrigid case is to rewrite FE so that it involves

an integral over a carefully chosen contour in Cd. For z = x + iy E Cd and w

(Cm_d let

Since If(z, w,

be the contour parameterized by the map

is an entire function of

we

can use Cauchy's theorem to

show

FE(z,w) =

J

is close to the copy of Rd Note that since Oh(0, 0)/Ox = 0, the contour = 0}. given by For the same reason, we have

for •

11

.Oh(x,w) Ox

provided

_i\t )

(x, w) is sufficiently close to the origin. The presence of the term

= r} appearing in means that the mtegrals over the side contours the change of contour process disappear as R oo. The reason we need this change of contour will become clear at a later time. Note in the rigid case, nothing has changed (since Oh(x, w)/Ox = 0). Now, we make the analogous change of contour as in the proof of Theorem 1 in Section 16.1. That is, we set

where we have used the notation

The Fourier integral approach in the nonrigid case

253

is well defined and holomorphic in E Cd for provided The expression > After dropping the hat and changing the contour of integration, as done in the proof of Theorem 1 in Section 16.1, we obtain

F€(z,w)

If(z, w,

= (27r)_d

= + presence of the term

=

f

f g(z')f(z'

—

z',

z'), we can restrict z and z' so that The means that the integrals over the side contours appearing in the change of contour process disappear ascc. 11 R} {Im As mentioned earlier, FE(z, w) is holomorphic in z E Cd and f on M as Since

—

I

0.

The definition of the hypoanalytic wave front set given in Definition 1 in Section 16.2 is unchanged for the nonrigid case except that the cone F must be a conical neighborhood of in Cd rather than an Rd.conical neighborhood as stated. The reason for this is that If(z, w, involves an integral over which is contained in Cd rather than an integral over Rd as in the rigid case. Theorem 1 in Section 16.2 holds without change in the nonrigid case. The basic ideas of the proof are similar to the ideas given in the rigid case except one must use the new transform If(z, w, defined above. For example, let us outline the key steps in the proof of the (a) implies (b) part of this theorem for the nonrigid case. Here, we are assuming that the hypoanalytic wave front set of a given CR function f is contained in a cone F and we are to show that f holomorphically extends to a set of the form w+{Fi } where F1 is a smaller subcone of the polar of F. As in the proof for the rigid case, the key idea is to estimate the real part of the exponent q(z, z', appearing in the definition of If(z, w, where z' = x' + ih(x', w) z = x + i(h(x, w) + v) (where v belongs to F1) and = ([1+ To carry out E this estimate, we shall require that there are no second-order pure terms in the Taylor expansion of h. In particular, we need for

We have z—

=x

—

x' + i(h(x, w) — h(x', w) + v)

= I+z.Oh(x,w)]j(x—x)+o(Ix—xI'2 )+iv. /

The notation o(t') for j 0 indicates terms that are small in absolute value

Further Results

254

Taking the Euclidean dot product of z — z' with

when compared to

([I + i(ah(x,

=

we obtain

(C) E

+o(Ix_x'l2)ICI+O(Ixl+ lwI)IvIICl+O(lvl2lCI). Therefore

Re{i(z — z')

—

—

<

-

-v

-

—v

+ C((lxI + lwI)lvI + vl2)lCl

+ C((IxI + lwI)Ivl + (1)

provided z, z', and w are contained in a suitably small neighborhood of the

origin. This is analogous to the estimate on the exponent given at the end of the proof of Lemma 3 in Section 16.1 for the rigid case. It is here that the reason for introducing the contour E } becomes apparent. For if we compute E Rd as we did for the rigid case, then we encounter the (z — z') for following additional error term:

Oh(x,w)]

j(x_x).C.

cannot absorb this error term by the expression Cl lx — x' 12 in the way we absorbed the term o(lx — x'I2lCl) to obtain (1). If f is a function of class Cd+2 on M, then we can integrate by parts as done in the proof of Lemma 3 in Section 16.1 to show that If(z, w, C) is dominated by an integrable function of C provided w and z are close to the origin and z = x + iy with y E r1. After letting f 0, we conclude that I is the boundary values of a function (z, w) '—' F(z, w) which is holomorphic in z provided (z, w) belongs to a sufficiently small neighborhood of the origin in M + r1. If I is CR, then F is holomorphic in w as well, as shown in Lemma 1 of Section 16.2. The proof of the converse of this theorem and the proof of Theorem 1 in We

Section 16.3 are similar to the proofs given for the rigid case after making suitable modifications as above.

17.2

The holomorphic extension of CR distributions

First, we mention that the approximation theorem (Theorem i in Chapter 13) holds for other classes of CR functions. For example, if f is a CR function of class C" then the approximation theorem produces a sequence of entire functions that converges to I in the topology of The analogous statement holds true

The holonwrphic extension of CR distributions

255

for the class of CR functions, for 1 p oo. The proofs of these facts are easy modifications of the proof given for the case of the sup-norm. The approximation theorem for the class of CR distributions is obtained by dualizmg the proof of the approximation theorem for the class of smooth CR

functions. To carry out the details (and the details for the remainder of the proofs of the theorems in Part III for CR distributions), it will be necessary to restrict the given CR distribution to totally real slices of the CR manifold M. Let M = {y = h(x,w)}. We identify a given function (or distribution) 1: M C

with the function/i: Rd x Cn_d

C, where fi(x,w) = f(x+ih(x,w),w).

It is possible to view a given CR distribution as a smooth (C°° ) map from the set {w E C.n_d} to the space of distributions on R" (in the x-variables). This means that for a given smooth, compactly supported function : C, the map w

= is a smooth function of w. To see this, we use Theorem 3 from Section 13.2 to write the projection of the tangential Cauchy—Riemann vector fields onto

= with

Mk = where

is the (1, k)th entry of

.OhV' 11+2—

I

Ox

is a CR distribution, then If = 0 and we can trade w-derivatives for x-derivatives. Together with a bootstrap argument, this shows that c1 is smooth. The CR extension theorem (Theorem 1 in Section 14.2) can now be generalized to include the class of CR distributions. However, we need to explain what is meant by stating that a CR distribution is the boundary values of a holomorphic

function F defined on a wedge M + if. For the rigid case, M = {y = h(w)}, we define the boundary values of F (denoted bF) by (bF(x, w),

= lim

f F(x+i(h(w) +tv),

w)dxdv(w) (1)

= 1. For the nonrigid case, E D(R'2 x Cn_d) where v E I', with F(x + i(h(w) + tv),w) is replaced by F(Z(x + itv,w),w) where for each w, Z(.,w) : Cd Cd is a smooth function with Z(x,w) = x + ih(x,w) for

for x E R" and where Z(z, w) satisfies the Cauchy—Riemann equations on

Further Results

256

Imz =

to infinite order. Suppose F has polynomial growth (this means IF(Z(x + itv, w), w)I Ct—N, for some positive constants C and N). It is shown in [BCI] that bF is a well-defined CR distribution on M which is independent of v E r. For the proof, the main idea is to show that the integral on the right side of (1) and all of its t-derivatives are bounded by Ct—N. The constant C is allowed to depend on the number of t-derivatives but N is not allowed to depend on the number of t-derivatives. This is accomplished by taking t-derivatives of the integral on the right side of (1); using the Cauchy— Riemann equations to turn them into x-derivatives, and then integrating by parts to put the x-derivatives Onto the function Repeated use of the Fundamental 0

Theorem of Calculus shows that if a function of t, t 0 and all of its tderivatives are bounded by Ct_N, then this function is continuous (in fact smooth) up to t = 0. The converse of the above boundary value result is the CR extension theorem in the distributional category: under the same hypothesis on M and using the same notation as in Theorem 1 of Section 14.2, for a given CR distribution f on M, there is a holomorphic function F on } with polynomial + i{fi fl growth such that bF = on A short cut for the proof of this theorem can be obtained by using the following fact (see [BR2]): for a given CR distribution there is an positive integer N and a CR function which is smooth (say C4) such that

f

d

N

{J1}

Mk is the vector field involving 0/Ox1 . defined above. In the we rigid case, we have Mk = O/Oxk. To extend the given CR distribution first extend the smooth CR function to a holomorphic function F1 (which we already know how to do) and then note that the CR distribution is the N boundary values of the holomorphic function For the ) { F1 }. nonrigid case, we refer the reader to [BR2I. In the Fourier transform proof of the CR extension theorem given in Chapter 16, we require the CR function f to be of class Under this smoothness assumption, we show (in the rigid case) that the transform If(z, w, is an integrable function of E Rd. This allows us to use the dominated convergence theorem to get rid of the annoying factor of appearing in Lemma 2 of Section 16.1 and thus simplify the proof of the CR extension theorem. This results in an apparent loss of derivatives. That is, if f is a CR function of class then its holomorphic extension appears to only be of class C' up to M. However, by using a little potential theory, Baouendi, Jacobowitz, and Treves have shown in Lemma 2.4 in EBJT) that there is no loss of derivatives. In where

. .

particular, the holomorphic extension of a continuous CR function is continuous

up to M.

CR extension near points of higher type

17.3

257

CR extension near points of higher type

If the convex hull of the image of the Levi form has empty interior, then CR extension to some open set in C" may be impossible. For example, CR extension to a fixed open set from the submanifold M = { (z1, z2, w1, w2) E C4; Imz1 = 0, = 1w112} is impossible (because M is contained in the plane {Imzi = 0}). If the convex hull of the image of the Levi form has empty interior, then higher order conditions are required to ensure CR extension to an open set in C". For a long time, the best theorem available in this context is a result of Baouendi and Rothschild (BR2I which handles the case of a point p of finite type in a semirigid CR submanifold, M (for the definitions of finite type and semirigid, see Chapter 12). Under these conditions, their theorem states that if is an open subset of M that contains p. then there is an open wedge Wr in C" such that all CR functions on w holomorphically extend to Wr. By

definition, awedgeinCT' isanopensetoftheform Wr =W'+{rnB(o,f)} iS an open subset of M which contains p and where r is an open convex cone in the normal space of M at p. The simplest example that illustrates this theorem is the codimension two submanifold M = {(z1, z2, w) E C3; Imz1 = wi2, Imz2 = iwl2Rew}. This manifold is rigid (and hence semirigid). The image of the Levi form at the origin is the ray {Imzi 0, = 0} which has empty interior in the two-dimensional normal space of M at the origin. So the CR extension theorem in Section 14.2 does not apply. In fact, the hypothesis of the CR extension theorem in Section 14.2 is never satisfied by a codimension two submanifold in C3 (because the holomorphic tangent space of such a manifold is one-dimensional and so the image of the Levi form is always a one-dimensional ray in the two-dimensional normal space). In the above example, the origin is a point of type (2,3) and so Baouendi and Rothschild's result states that CR functions near the origin in M holomorphically extend to a wedge in C3 Recently, Tumanov m has significantly strengthened the above result. Not only has he shown that the above semirigid condition is unnecessary but he has also shown that finite type is unnecessary in certain situations. A CR structure (M, L) is minimal at a point p E M if there does not exist a proper submanifold

where w'

NofMwhichcontainspsuchthatLiN cTC(N). IfapointpE Misapoint of finite type (i.e., all the Hörmander numbers are finite) then M is minimal at p. This is because the existence of a proper submanifold N with LIN C Tc (N) means that all the Lie brackets of all lengths of vector fields from L L at p must be contained in Tc (N) (since Tc (N) is involutive). So any vector in (N) cannot be realized as a Lie bracket of any order (M) that is not in

atpofvectorfields in

sopis notapointof finite type. IfM isreal

analytic, then minimality is equivalent to finite type (see [BR3]). Tumanov has shown that if a generic smooth CR submanifold M of C" is minimal at p. then

CR functions on M near p holomorphically extend to some wedge in C". In

258

Further Results

addition, Baouendi and Rothschild [BR3] have shown that the converse holds, that is, if CR functions on M near p E M holomorphically extend to a wedge in C'2, then M must be minimal at p. We should mention that these necessary and sufficient conditions for the case of a real hypersurface had previously been discovered by Trepreau [Tr]. The reader should not get the impression that the above-mentioned results of Tumanov and Baouendi and Rothschild are the last word on CR extension. There are many interesting harmonic analysis problems remaining in the field of CR extension. For example, if a CR function is locally integrable, then what can be said about pointwise limits (on M) of its holomorphic extension? There is extensive literature on the approach regions allowed for pointwise limits of holomorphic functions for a domain bounded by a real hypersurface (see [Sti or [BDN]). It would be interesting to obtain analogous results for wedges in C'2 where the edge of the wedge is a higher codimension submanifold of the type we have been discussing (i.e., finite type or minimal). Some results along these lines for points where all the Hörmander numbers are 2 will be forthcoming in [BN}. In the case where each Hörmander number of a point p E M is 2, a precise

description of the wedge Wr' is given in the CR extension theorem in Section 14.2. Less can be said of the geometry and size of the wedge if the point p has higher type. However, a few observations can be noted. First, the wedge always points in the direction(s) in the normal space which can be generated by (J of) the Lie brackets of Hc (M) of lowest order. In the example above, IwI2Rew}, the wedge is cenlmz2 M = {(zI,z2,w) E C3; Imz1 = tered around the positive Imz1 axis since this is the direction that is in the image

of the Levi form at the origin. The second observation is that the size of the wedge is governed by the radius of the set w C M on which the CR functions are defined and by the HOrmander numbers (mi,.. . md) which make up the type of the point p. If w contains a Euclidean ball of radius > 0 in M, then ,

the dimensions of the normal cross section of the wedge at p are proportional to 45m, The following figure illustrates the case where (m1 m2) = (2,3). This result can be seen by the following scaling argument. Suppose the point ,

p E M is the origin and the defining functions for M are put in the Bloom— Graham normal form (see Section 12.1) I/i = Pmi (w) + e1 (x, w)

Y2 =pm2(x1,w)+e2(x,w)

Yd

pmd(X1,...,Xd_1,W)+ed(X,W)

is a polynomial of weight m3 and where e2 is a smooth function of C'2 given by weight greater than rn3. Consider the biholomorphism H5 : C'2 H5(z,w) = ,ömd2d,45W). Let be an open set in M that contains where

CR extension near points of higher type

259

p

M

FIGURE 17.1

a Euclidean ball of radius 5 in M about the origin. Let and M5 be the contains image of and M under the map Hi', respectively. Note that a ball of radius 1 in M5 about the origin. The terms of lowest weight in the defining equations for M5 are the same as the terms of lowest weight for the defining equations for M. Moreover, the terms of higher weight converge to zero as 6 '—i 0. Therefore, the above-mentioned result of Tumanov shows that that CR functions on C M6 holomorphically extend to some wedge in = H5 } C M is independent of (5. It follows that all CR functions on holomorphically extend to a wedge with the dimensions described above. This argument works just as well if we only assume contains a nonisotropic ball of radius 6 of the type defined by Nagel, Stein, and Wainger (see [NSWJ). Up to proportionality constants, the above size estimate on the wedge is sharp. That is, we cannot expect to holomorphically extend CR functions to a larger wedge. For example, consider M = {(z1 , w) e C3; Yi = wi2, Y2 = be the intersection of M with the Euclidean ball iwI2Rew}. For 6 > 0, let of radius 6 in C3 centered at the origin. Also let r5 be the cone in R2 given by is contained in the wedge öyi}. The convex hull of the set {0 62}}. This wedge has the dimensions described above + {r5 fl {0 y' for a point (in this case, the origin) of type (2,3). Since CR functions on a set cannot (in general) be holomorphically extended past its convex hull, the set + {r5 fl {0 y' 62}} contains the largest set to which all CR functions on

holomorphically extend.

260

Further Results

Finally we mention that even if a pomi p E M does not satisfy the hypothesis of Tumanov 's result mentioned above, CR functions on M may locally extend to CR functions on a CR manifold of higher dimension (rather than to holomorphic functions on an open set in Ca). Suppose the type of the point p is (mi,..., md) with m1 ... If m1 is finite (the other are allowed to be infinite),

then it can be shown that near p. M is the boundary of a manifold Al with dirnR(M) = dimR(M) + 1 such that CR functions on M locally extend to CR functions on M. If in1 = 2, then the Levi form of Jt'f at p is not identically zero and this result is due to Hill and Taiani [HTI]. The case rn.1 2 is handled in [BPIJ.

17.4

Analytic hypoeflipticity

Suppose

...

,

are vector fields with real analytic, complex-valued co-

efficients defined on a real analytic manifold M. We say that the system {L1,. .. , is analytic hypoelliptic near a point p Al if whenever real analytic functions near p and whenever u is a distributional solution to the equations . .

,

are

(1)

near p in M, then a is also real analytic near p in M. Let L be the subbundle of Tc(M) generated by L1,.. .,Lm. If L is involutive, then (M,L) is a real analytic CR structure which (according to Theorem 1 in Chapter 11) we can imbed into In this case, the real analytic hypoellipticity of the system near a point p M is equivalent to showing that the each CR distribution near p in M is real analytic near p. This equivalency is seen as is analytic hypoelliptic at follows. It is clear that if the system L1. . p, then each CR distribution near p must be real analytic. For the converse, .

,

suppose (1) has a solution a. We can use the Cauchy—Kowalev sky theorem to locally solve the system of equations (1) with a real analytic solution UI. The function a — u1 is a CR function and u is real analytic if and only if u — is real analytic. The converse now follows. If M is such that each CR distribution near a point p E M holomorphically which contains p, then each CR distribution extends to a neighborhood in

near p must be real analytic and thus by the discussion above, the system = H°" (M) is real analytic hypoelliptic at p. The converse also holds and this is shown in Section 5 of (Bill. There are a number of situations where CR functions near a point p in M which contains a given locally holomorphically extend to a neighborhood in point p. By the above discussion, this means that the resulting system of tangential Cauchy—Riemann equations is analytic hypoelliptic at p. The first situation is the one described in Theorems 2 and 3 in Section 14.2. Here, the convex hull

Analytic hypoeWpticity

261

of the image of the Levi form at the given point p is the entire normal space of M at p. This situation also occurs if the Levi form at p vanishes identically, but the commutators of length at most 3 at p generated from HC (M) span (over R). This result is contained in [B]. For hypersurfaces, a result of Baouendi and Treves [BT3] states that if p is a point of odd type, then CR functions on M near p locally holomorphically extend to both sides of M. Necessary and sufficient conditions real analytic hypoellipticity have been

found by Baouendi and Treves in the tube like case. A submanifold M = is said to be tube like if the graphing function h depends {y = h(x, w)} of only on u = Rew. As usual, we are using (z, w) as coordinates for where z = x + iy E Cd and w = u + iv E In this case, (d = Baouendi and Treves [BT2] have shown that the system of tangential Cauchy— Riemann equations for M is analytic hypoelliptic at the origin if and only if for each vector in the normal space of M at 0 (No(M) Re'), the origin is not a local minimum of the map u h(u) u E

There is also a considerable amount of literature on the related problem of deciding when a smooth CR map between two real analytic CR manifolds must be real analytic. The simplest case to consider is the case of two real analytic hypersurfaces M and M' in and a smooth (say C°°) CR diffeomorphism 4): M '—' M'. The theorem in this context is the following: if the Levi form

of M at p is nondegenerate (i.e., the matrix representing the Levi form at p has maximal rank) then any smooth CR diffeomorphism from a neighborhood

of p in M to M' must be real analytic near p. Of course, if the Levi form of M at p has eigenvalues of opposite sign, then every CR function near p holomorphically extends to both sides of M and therefore must be real analytic

as discussed above. However, if M is strictly pseudoconvex at p. then CR functions only extend to one side of Al and so some new ideas are needed to prove this theorem. One approach, taken in [L3] or [P1, uses a reflection principle. Another approach, taken in [BiT], uses CR extension results. Roughly, the latter approach is the following. Since the Levi form of M at p is not identically zero the component functions of a CR map 4) (which are CR functions) must holomorphically extend to one side of M. Therefore, the conjugates of the component functions of 4) must holomorphically extend to the other side of Al. Since the Levi form of Al at p has maximal rank and since 4) is a CR diffeomorphism, the Levi form of M' at 4)(p) must also have maximal rank. This fact together with the implicit function theorem and the fact that the defining functions for M' are real analytic can be used to show that the component functions of 4) are holomorphic functions of the conjugates of the component functions of 4) and their derivatives. Since the conjugates of the component functions of 4) holomorphically extend to the other side of M, one obtains that the component functions of 4) holomorphically extend to the other side of M as well, and hence the component functions of 4) holomorphically extend to both sides of M. Therefore, 4) is real analytic on M near p, as desired. l'his theorem and its proof generalize to higher codimension as shown in

262

Further Results

[Bill. Assume the convex hull of the image of the Levi form of M at p E M has nonempty interior as in the statement of our CR extension theorem. So the components of a CR map 4' holomorphically extend to a wedge of the form Wr = + zr as in the conclusion of our CR extension theorem. The conjugates of the component functions of 4' must therefore holomorphically extend to a wedge of the form W_r = w — ii'. A nondegeneracy condition on the Levi form given in Section 3 of [BiT] and the implicit function theorem can be used to show that the component functions of 4' are holomorphic functions of the conjugates of the component functions of 4' and their derivatives. Thus, 4' holomorphically extends to the union of the wedges Wi- U W_ r. By the edge of the wedge theorem (see [Ro]), 4' holomorphically extends to a neighborhood of p in and therefore 4' is real analytic near p, as desired. Baouendi, Jacobowitz, and Treves have generalized these theorems to handle higher order situations (i.e., cases where the Levi forms vanish at a point). Results along these lines can be found in [BiT] and [BR 1] (see also the references given in these papers). We also define the notion of C°° hypoellipticity by replacing real analytic with C°° in the above definition of real analytic hypoellipticity. C°° hypoellipticity is a more difficult concept to analyze than real analytic hypoellipticity. For one thing, the Cauchy—Kowalevsky theorem does not hold for the C°° category (see Chapter 23, where Hans Lewy's example is discussed). Also, Maire [M] has found an example of a system of first-order vector fields with real analytic coefficients which is not C°° hypoelliptic. This example is tube like and satisfies the hypothesis of the above-mentioned result for tube like CR structures (see [BT2J) and so this system of vector fields is real analytic hypoelliptic. As far as positive results concerning C°° hypoellipticity, we mention a result of Shaw [Shi]. Her result handles the case of a set of vector fields L = L1,. . with smooth coefficients defined on an open subset of RN. Unlike the results mentioned above, she does not need to assume that L is involutive. She shows that if p is a point in such that (M)/L L) is the convex hull of the image of the Levi form of L at p, the system L is hypoelliptic at p. .

Part IV Solvability of the Tangential Cauchy—Riemann Complex

In Part II, we defined a complex to be a collection of vector spaces {Aq, q O} together with maps dq: Aq Aq+i with the property that dq+i 0 dq = 0. We = have discussed three classes of complexes: the first is where — the space of smooth q-forms on a manifold M — with dq = dM — the

exterior derivative on M; the second is where M is a complex manifold and Aq = (p fixed) and dq = 0 — the Cauchy—Riemann operator on M; the third is where M is a CR manifold and Aq = and dq = — the

tangential Cauchy—Riemann operator on M. For any complex {

Aq

Aq+,; q 0}, a natural question to ask is whether or not it is solvable. That is, if f E Aq with dqf = 0, does there exist u E Aq_i with dq_iu = 1. For our three classes of complexes, this solvability question can be posed locally or globally. In the global problem, say for the tangential Cauchy—Riemann operator, we assume that f E with 0Mf = Oon all of M and then we ask whether or not there exists u E with OMu = f on all of M. In the local question, we ask whether or not there is a local neighborhood basis {WA;

)..

> 0) about any given point p M such that for each f E

f

with OMf = 0 on WA there is an element u with OMU = on In all three classes of complexes, the answer to either the global or the local question does not automatically yield the answer to the other. For example, we cannot use the local solvability of 0M together with a partition of unity argument to obtain the global solvability of '9M because the product of a 0M -closed form with a smooth cutoff function is typically no longer t9M -closed. For the same reason, we cannot deduce local solvability from global solvability. For the exterior derivative and the Cauchy—Riemann operators, the answer to the solvability question is well understood. Both complexes can always be solved locally. The global solvability of the exterior derivative depends on global topological conditions on the manifold (the cohomology groups). From the theory of several complex variables, the global solvability of the Cauchy—Riemann

263

264

Solvability of the Tangential Cauchy—Riemann Complex

operator depends on the holomorphic convexity of the complex manifold. Much less is known about the global and local solvability of the tangential Cauchy—Riemann operator on a CR manifold Al. However, the theory is pretty well understood if M is a strictly pseudoconvex hypersurface in and this is the subject of much of Part IV of this book. We show that the tangential Cauchy— Riemann complex is both locally and globally solvable except at top degree. For 1 q < n — 1. the equation 8M = f for f E ( M) is overdetermined (more equations than unknown functions), and the condition = 0 is the correct compatibility condition required for both local and global solvability. If q = n — (top degree), the equation u = is no longer overdetermined.

f

1

E e°"'(M) then the equation

In fact, if u E

I

= consists of only one first-order partial differential equation. Based upon one's experience with d and 0, one would expect that the tangential Cauchy—Riemann

complex should at least be locally solvable at top degree. In fact, if Al and f are real analytic then local solvability follows from the Cauchy—Kowalevsky theorem. The surprise (provided by Hans Lewy's nonsolvability example [L2]) is that local solvability does not necessarily hold if f is only smooth. Adding the condition OM = 0 does not help matters since this condition always holds if f has top degree. Instead, there are other criterion for both local and global solvability at top degree. which are discussed toward the end of Part IV. We employ the integral kernel approach of Henkin [Hell, [He21 for the so-

f

lution of the tangential Cauchy—Riemann complex. We shall not discuss the £2 technique of Horrnander, Kohn et al. because there are ample references for this approach (see Folland and Kohn's book [FK]). Aside from the aesthetic appeal of exhibiting an explicit integral kernel formula for the solution to the tangential Cauchy—Riemann equations, the kernel approach makes estimating the solution rather easy (at least in the global case).

Part IV is organized as follows. In Chapter 18, we introduce the calculus of kernels and we define the concept of a fundamental solution for d, 3, and Various fundamental solutions for d and 3 are discussed in Chapter 19. As an application we prove Bochner's global CR extension theorem In Chapter 20, [Boc] for the boundary of a smooth bounded domain in we introduce Henkin's kernels, which along with the fundamental solution for O form the building blocks for the integral kernel solution to the tangential Cauchy—Riemann operator. Instead of using Henkin's notation, we use a more streamlined notation due to Harvey and Polking [HP]. Two global fundamenare introduced in Chapter 21 along with a criterion for tal solutions to global solvability at top degree. In Chapter 22, one of these global fundamental

solutions is modified to yield a solution to the local problem. Hans Lewy's local nonsolvability example is given in Chapter. along with a more general criterion of Henkin's [He2) for local solvability.

18 Kernel Calculus

Our presentation in this chapter closely follows Section 1 in [HP].

18.1

Definitions

To motivate our discussion of kernels, let K be a smooth form on RN' x RN

N+N'). We can view K as an operator from VP(R")

of degree q (O q

by defining

to

f

XERN.

yEW"

Here, it is understood that the only contributing term to this integral is the containing dy = dy1 A ... A dyN'. The dx's component of K(x, y) A appearing in this component are moved to the right (outside the integral) and then the y and dy are integrated. Since the degree of K(y, x) A is p + q, (Kcb)(x) is a differential form in x of degree q + p — N'. Recall (see Chapter 6) that as a current on RN, the form K(q5) acts on elements (with s = N + N' — (q +p)) by integration. So if E D3(RN), then

of

=

j(j VERN'

= The term (K,

0

XRN

is

well defined for any current K E x We can also replace RN and RN'

RN) and any q5 E VP(RN'), with oriented manifolds X and V of dimensions N and N', respectively. In this case, the manifold Y x X has an orientation induced on it by Y and X.

265

Kernel Calculus

266

= (xii... ,xN): U —+ RN are orientation-preserving coordinate charts for X and Y, respectively, then dy1 A...AdYN' Ads1 A...AdXN determines an orientation for Y xX in the RN' and

That is, if Y = (yi,... ,yN'): V

sense described in Section 2.5. The above discussion motivates the following definition.

(Y x X) is called a kernel of degree q on A current K in 1yq+p.- N (X) Y x X. This kernel can be regarded as an operator K: VP(Y)

DEFINITION I by

for

E VP(Y) and

Note that if

E 1)8(X) with 8 = N + N' — (q + p).

—p

in Vt(YxX); therefore,

in

This shows that

is a well-defined current

on X. If q = N' + r, then we say that K is a kernel of type r. In this case, K is an operator from VP(Y) to 7YP+T(X). For our applications r will usually be 0,—I, or —2.

Example I As already mentioned, any smooth form K on Y x X defines a kernel. Many of the theorems about kernels are motivated by considering this class of kernels.

Another closely related class of kernels is the space of forms on Y x X with locally integrable coefficients.

1]

Example 2

Y be an oriented smooth manifold of dimension N. Let = Let X given by in{(x, x); x E X} be the diagonal of X x X. The current is N. tegration over the diagonal is a kernel of type 0 since the degree of D'P(X) represents the identity map. For if VP(X) As an operator, [s]: E D"(X), then

A parameterization for

is given by x '.—. (x, x), x E X. We have

f

rEX

=

Definitions

267

= as claimed. This is analogous to the fact that the operation of convolution with the delta function is the identity operator (see Chapter 5). This analogy is made even clearer by writing the current as a form with distribution coefficients So,

=6o(x—y)d(xi —yl)A...Ad(XN Example 3 Let X and Y be oriented manifolds and suppose f: X Let Gr{f} be the graph of I in Yx X, i.e.,

—YN).

—,

I]

Y is a smooth map.

Gr{f} = {(y,x); y = f(x)}.

Gr{f} given by F(x) = (f(x), x) is a global parameterization for Gr{f}. Let us orient Gr{f} by pushing the orientation on X to Gr{f} via F. That is, the collection C T{Qr{f}} is said to be

The map F: X —÷

. .. , positively oriented if and only if is positively oriented on X. With this orientation on Gr{f}, the map F is orientation preserving. The current (or kernel) [Gr{f}] on Y x X has dimension N and therefore degree N' (where N = dimX and N' = dimY). As an operator from DP(Y) to D'P(X), we claim [Cr{f}] represents the pull back map f. To see this, let E DP(Y) and E then

= = Gr{f} Since

F: X

Gr{f}

is orientation preserving, we have

([Gr{f}](Ø),

=

f

®

= Therefore, as claimed. Note that if X = Y and f(x) = = U x E X, then this example reduces to Example 2 above.

Now let us discuss the adjoin: of a kernel K denoted by K'. If K is an operator from V(Y) to V"(X), then K' is an operator from V(X) to V"(Y) and it is defined by

=

E V(Y),

V(X).

268

Kernel Calculus

If K is of type r then K' is of type N' — N + r. We wish to find a convenient way of computing K' from K. To motivate the calculation, we first suppose that K is a smooth form on Y x X. If E D*(Y) and E V*(X), then

= =

=

f J K(y, x) A

A

xEX y€Y

± J 0(y) A ( yEY

f

K(y, x) A

vEX

J K(y,x) A

=±

xEX

'yEY

The sign in front depends on the degrees of the forms involved and it will be resolved later. From the above calculation, we have

=

±

f K(y,x) A

x that the variable of integration in K(ç5)(x)). If X = Y, then switching x with y shows that K'(y, x) = ±K(x, y). Switching x and y means that dx3 is also switched with dy3.

To generalize this calculation for more general kernels, we introduce the switch map s: X x Y Y x X, s(x, y) = (y, x). Formally, the pull back s is the identity map on forms, i.e., for e V(Y) = 0(u) A and E D(X). However in local coordinates (x,y) for X x Y, we have

Det(Ds) = (_1)NN' changes the orientation by the factor (_l)NN'. For X), we obtain

Hence,

f

YxX

=

E

f

XxY

This formula extends to currents on Y x X. Since S is a diffeomorphism, s*K is well defined for K TYt(Y x X) (see Definition 2 in Section 6.2). In addition, 84 = for 4 V*(Y xX) in view of the remark after

Definitions

Lemma

269

in Section 6.2. We obtain

2

= (s_I os*K,4,)yxx = —i

\S

—

sr.-'

WIXxY.

This equation is the analogue of (1) for the pairing between currents and smooth forms. Now we compute the kernel of

K' and keep careful track of the minus signs. VP(Y) and x X) is a kernel of type r. Let 0

Suppose K

We have

= =

Now, s(0® and N

—

p

—

(by (2)).

A In addition, the degree of = r, respectively. Commuting 0(u) and yields

= The degree of and

and

isp

Ø)v

is N'

p and the degree of

—

is p. Commuting

yields

= (_

l)NN'+P(N+N'_r)(Ø,

So we have

= To simplify the notation, we introduce the map c:

cT = Since the degree of We obtain

for is

N' —p,

T

—

E

we have

= (_I)NN'CN+N'_rS*K Define the transpose of K, Kt E

x Y), by

Kt = We have proved the following.

LEMMA I

Suppose X and Y are smooth, oriented manifolds of dimensions N and N', x X)). respectively. Suppose K is a current of type r (i.e., K E Then

K'

CN+Nr{Kt}.

Kernel Calculus

270

Kt is easy to compute, this lemma gives a convenient formula for the adjoint, K'. It may happen that a kernel sends smooth, compactly supported forms to smooth forms (rather than to currents with nonsmooth coefficients). We single out this special class of kernels. Since

DEFINITION 2

Suppose K E DIN' +r (Y x X).

K is called a

(a)

If K is a continuous map from 1Y(Y) to EP-'-"(X),

(b)

regular kernel. JfK extends to a continuous operator from E'P(Y) to is called an extendable kernel.

(c)

If both K and K' are regular kernels, then K is called biregular.

then

then

K

LEMMA 2

if a kernel K E V1N'.Ft.(Y PROOF

x X) is regular, then K' is extendable.

If T E E'(X), then define

(Ø,K'(T))y = (K(Ø),T)x,

for

E D'(Y).

E e(X)). If The right side is well defined because K is regular (so K(Ø) in e'(X); therefore in D(Y), then by hypothesis T)x. l'his shows that K'(T) is a well-defined D'T)x —' T in current on Y. In a similar manner, the reader can show that if ef*(X) then K' is extendable, as desired. K'(T) I

The lemma implies that if K is biregular, then both K and K' are extendable.

An important class of biregular kernels is the class of kernels of convolution RN be defined by type, which we now describe. Let r: RN x

r(y,x) = x — y. (RN) The linear map r has maximal rank and therefore the pull back r*: D' (RN x IRN) is well defined (see Definition 2 and Lemma 5 in Section 6.2). x RN) is said to be of convolution type A kernel K V' if there exists a current k DII(RN) with K = rk.

DEFINITION 3

then the kernel K(y,x) = So, for example, if k = udy' with u E is a kernel of convolution type. y)d(x (rk)(y, x) = u(x — — y)' LEMMA 3

A kernel of convolution type is biregular.

Definitions

271

(RN), and

(Rh'),

Suppose k the definitions, we have

PROOF

(r*k(th),

E

(q +p)

= (r*k, th ® ?,L')RN xRN

=

(3)

Suppose k = udx1 with u E 2)' (1W"), and suppose 6 = 9, E V(RN). From Lemma I in Section 6.2, we obtain

=

(RN) From

f

g(y)h(x +

=

with

A d(x +

where the only nontrivial contribution to this integral comes from the terms involving dy = dy1 A .. dyN. Depending on the multiindices J and J', the coefficient of the form is either 0 or the following function of x (up ® to a + or — sign) .

f g(y)h(x + y)dy

=

f

g(y —

= = g(—t). Together with (3), this means that 0 or it is the term where

V5)aN is either

=

up to a ± or — sign. So the coefficient function of (rk)(4) is either 0 or g, up to a + or — sign which depends only on the indices I, J, and J'. If u E 1Y(RN), then the operator g u * g for g E D(JW') is a continuous linear map from D(RN) to (see Lemma 1 in Section 5.2). Therefore,

K=

T*k is regular.

If K = T*k, then we leave it to the reader to show that Kt = where k v*k and v: RN is defined by v(x) —x. So Kt is also a kernel of convolution type. It follows that Kt is regular, and so K is biregular, as desired. I The above definitions of kernels, type, regularity, etc., also apply to the case

when X and Y are complex manifolds or CR manifolds. The only difference is that we must keep track of bidegrees rather than just degrees. For example, if X and Y are complex manifolds of complex dimension n. and m respectively, a

current ofbidegree (m+r,m+s) onYx Xis said to be a kernel of type (r, s) and K can be regarded as an operator from VPQ(Y) to Since a complex manifold has even real dimension, many of the minus signs in the above discussion disappear. For example, Kt = S*K in the complex manifold setting.

Kernel Calculus

272

For CR manifolds, the bidegree counting is a little more complicated. Suppose

M and M' are CR manifolds. Suppose dimR M = 2m + d and dimR M' = 2m' + d' where d and d' are the CR codimensions of M and M', respectively. A form of top degree on M' has bidegree (m' + d', m') (see Section 8.1 or 8.2).

Suppose K is a kernel of bidegree (m' + d' + r, m' + s) on M' >< M. Then K is said to have type (r, s) and K can be regarded as an operator from For most of our applications, M = M' will be a hypersurface in to which means m = m' = n — and d = d' = 1. In addition, r will usually 1

beOandswillbeOor—1.

18.2

A homotopy formula

In this section, we develop a homotopy formula for the exterior derivative, the Cauchy—Riemann operator and the tangential Cauchy—Riemann operator. Suppose X is a smooth manifold and let K E V1N+T(X x X). Consider the

current dK where d is the exterior derivative on X x X. Viewed as a kernel The homotopy formula operator, dK is a map from DP(X) to given in the next theorem relates this operator with K and the exterior derivative on X. The 0 and 0M versions are also given for the case of a complex manifold and CR manifold, respectively. THEOREM I HOMOTOPY FORMULA of real dimension N. JfK (a) Suppose X is a smooth

DtN+r(X x

X), then

as operators on D*(X). (b)

Suppose X is a complex manifold of dimension n. JfK E X), then

<

OK = as operators on V*(x) (c) Suppose M is a CR manifold of real dimension 2m + d where d is the CR then codimension of M. If K E OMxMK =

oK +

as operators on On the left sides of the equations in parts (a) and (b), d and 8 refer to the d and 0 operators on the product space X x X, whereas on the right sides, d and

A homotopy formula

273

0 refer to the d and 8 operators on X. The notation in part (c) for the tangential Cauchy—Riemann operator is more clear in this respect.

PROOF We shall prove part (a). Part (b) follows from part (a) by taking the piece of bidegree (n+r,n+s+l) of the equation in part (a). Likewise, part(c) follows by taking the piece of bidegree (m +r + d, m + s + 1) of the equation in part (a) and the intrinsic definition of the 0M operator (see Section 8.2). If K is a form with C' coefficients, then (a) reads

=

J

J

yEX

yEX

where

is the exterior derivative on X x X. The above equation is established

give the first term on the right side of the above equation. The factor of (—I) N results from commuting dx past N dy's so that all the dx's appear to the right of the y-integral as required by the convention set down at the beginning of this

chapter. The second term on the right is obtained by an integration by parts with d,,, and noting that K(y, x) has degree N + r (resulting in the factor of If K has distribution coefficients, then the argument is essentially the same. VP(X),,,b Suppose From Definition I in the previous section, we obtain

= =

(I)

l'he second equality follows from the definition of the exterior derivative of a current, and the product rule for d. We have

= and

(K(cb),di,b)x

= Inserting these two equations into (I) yields

= and the homotopy formula follows.

+ I

274

Kernel Calculus

Of special significance is the current equation

dK = the current given by integration over the diagonal = {(x, x); x E is the identity I, as shown in the previous section. Note that has degree N and so K has degree N — I. If K satisfies (2), then in view of Theorem 1, we have where

is

X}. This is because as an operator, [Lx]

doK+Kod= I as operators on D*(X). In particular, if

e DP(X) with

0 on X, then

d{K(ø)} = So

the equation du =

has a solution u = K(q). Analogous statements hold

for 0 on a complex manifold. This discussion motivates the following definition. DEFINITION current K E

I

X is a smooth manifold of real dimension N. A x X) that satisfies

(a) Suppose

on XxX is called a fundamental solution for d.

(b) Suppose X is a complex manifold of complex dimension n. A current K E Vhhl.n_t(X x X) that satisfies OK = is

called a fundamental solution for 0. The reader should note the analogy with the concept of a fundamental solution

T for a partial differential operator P(D) with constant coefficients (i.e., a solution to P(T) With_the d or 0 operators, takes the place of bo and a solution to du = or flu = 0 is then obtained by setting n = This is analogous to the solution to P(D)u = which is obtained by setting u = T * (see Section 5.4). At this point, the reader may wonder why we have not defined the concept of a fundamental solution for 0M for a CR manifold M. The reason for this is that except for very special M (such as a foliation of complex manifolds) no solution to the equation 0M MK = exists. However, if M is a strictly pseudoconvex hypersurface in Ci', we shall construct kernels, K, which solve this equation up to an error term which, as an operator, only acts nontrivially on forms of bottom and top degree. We shall then call such a kernel a fundamental solution for t9M.

275

A homolopy formuLa

If K is an extendable current that is a fundamental solution to d, then the equation

d(K(T)) + K(dT) = T

(3)

holds for currents in eI* (X). To see this, we approximate the given current E V(X), n = 1,2,... (see Theorem 1 in TE Section 22.2 and also Section 6.1). The exterior derivative is a continuous dT. Since operator on 7Y(X) (with the weak topology). Therefore K is extendable, —' K(T) and K(dT). Equation (3) now follows. Similar remarks hold for 0. For future reference, we summarize this discussion in the following theorem. THEOREM 2

(a) Suppose K is an extendable current that is a fundamental solution for d on a smooth oriented manifold X. If T E E't(X), then

d(K(T)) + K(dT) = T. (b) Suppose K is an extendable current that is a fundamental solution for 0 on a complex manifold X. If T E (X), then

+ K(OT) = T. Of particular interest is the current of degree I given by T = [OD}f where D C X is an open set with smooth boundary and f E D'(OD). If OD is given the usual boundary orientation (see Section 8.5), then dXD = —[OD] where XD is the characteristic function on D. If K is an extendable current, then K([OD] A f) is a well-defined current on X. An example of an extendable current is a differential form K with smooth coefficients on X x X. In this case K([OD] A f)(x) = (K(y, x), [OD] A (f)(y))VEX

=

A

f

K(y,x) A 1(y),

x E X.

yEOD

indicates that y is the "variable of integra, tion." This formula also holds for forms K(y, x) whose coefficients are locally integrable in y E OD. Suppose D cc X is a domain with smooth boundary and let f E E*(X). If K is a fundamental solution for d on X then by letting T = XDI in part (a) of Theorem 2, we obtain

As in Part I, the notation (

d{K(xDf)} + K(XDdf) — K([OD] A 1) = XDI.

276

Kernel Calculus

If df =

0 on D then the boundary integral term K([OD] A f) is the obstruction to solving the equation du f on D.

Similar remarks hold for 0 on a complex manifold X. Let D C X be a domain in X with smooth boundary. The equation dxD

= — [ODJ splits into two equations: '9XD = —[t9D]°" and OXD = —[OD}"°. If K((,z) is a form with locally integrable coefficients on OD, then for I

K([ODJ°" A f)(z)

A

f

[K((, z) A

z

X

(E8D

indicates the piece of bidegree (n,n —

where

in (. If K is a fundamental solution for 0 on X, then by letting T = xDf 1)

in

part (b) of Theorem 2, we obtain

+ K(XDC9f) — K([OD]°" A 1)

If Of =

0

XDI.

(4)

on D, then the boundary integral K([OD]°' A f) is the obstruction to

solving the equation On = f on D. In the next chapter, fundamental solutions for d and 0 will be constructed. In Chapter 20, additional kernels will be presented

which will allow us to solve the equation On = K([OD]°" A 1) provided D is strictly convex. This will enable us to solve the equation On = on a strictly convex domain D. The case where f E £P'°(X) deserves special note. If K E x X) then Kis of type (0, —1) and so K(xDf) 0. In simpler terms, n — d('s and n — d('s are required for integration on X, but K has at most (n — 1) — dc's and f has none; thus K(f) = 0. If Of = 0 on D, then (4) becomes

f

—K([OD]°" A f) = XDI.

In other words, we have

f(z)

=

f

[K((, z) A

for

z E D.

(E8D

This means that any fundamental solution for 0 on X reproduces holomorphic

functions on D. For future reference, we summarize this discussion in the following theorem. THEOREM 3

Let X be a complex manifold and suppose D cc X is an open set with smooth

boundary in X. Suppose K is an extendable kernel that is a fundamental solution for 8 on X. Suppose f is holomorphic on D and continuous on D. Then

XDf = —K([oDI°" A f).

19 Fundamental Solutions for the Exterior Derivative and Cauchy—Riemann Operators

In the previous chapter. we defined a fundamental solution for the exterior x to be a kernel K e derivative operator d: RN) which satisfies the equation

dK = where d on the left is the exterior derivative on RN x solution for the Cauchy—Riemann operator 0: analogous equation is

RZ%T. For

a fundamental the

In this chapter, we construct the ray kernel and the spherical kernel which are fundamental solutions for d on RN. We then construct the Cauchy kernel on a slice arid the Bochner—Martinelli kernel which are fundamental solutions for All of these kernels are convolution kernels and therefore they are 0 on biregular (and hence extendable to currents). Even though the fundamental solutions for the exterior derivative do not play a role in the construction of the solution to the tangential Cauchy—Riemann equations, we present them for three reasons. First, the fundamental solutions for d are easy to construct. Second, they are interesting in their own right. Third, it is interesting to see the parallels between the fundamental solutions for d and their counterparts for 0. Our approach is to first solve the equations

and

Ok=[O]

on

Here, [0] is the current of degree N on RN (or degree 2n on

given by

277

Exterior Derivative and Cauchy-Riemann

278

evaluation at the origin, i.e.,

([01,1) = 1(0)

for

I e DO(RN) (or

rk

where Our fundamental solution K for d (or 0) is then given by K = pN is defined by i-(y,x) = x y (or r((,z) = z — (for r: RN x RN — By definition, K is a kernel of convolution type. To see that K is (,z E a fundamental solution, we note that r*[01 = (ix] (see Lemma 6 in Section 6.2)

and note that Cs).

19.1

commutes with d (in the case of RN) and 0 (in the case of

Fundamental solutions for d on RN

We present two fundamental solutions for the exterior derivative.

The kernel on a ray The easiest solution to the equation

dk=(_l)N[0)

RN

is obtained by letting k be the current given by integration over a ray emanating

from the origin. Let o be a unit vector in Rh'. Let

= {tc;t 0}. Orient ka so that o is positively oriented. The current [ku] given by

a cuffent of dimension I and therefore degree N — I. By Theorem I in = (_l)N[0k0] = (.... 1)N[0] (this is just Stokes' theorem). Pulling back this equation via r where r(y, x) = x — y, we obtain is

Section 6.2,

d{r[k(,]} = Therefore, r' [ka) is a fundamental solution to d. This kernel is called the ray

kernel.

The spherical kernel and therefore its distriThe kernel [ku] defined above has support in the ray bution coefficients are not locally integrable functions. Sometimes it is desirable

Fundamental solutions/or d on Ri"

279

to have a fundamental solution with integrable coefficients. The spherical kernel

is such a kernel. To construct the spherical kernel, we start with a fundamental solution for on RN. Let

T(x) =

(2

—

N)_l(wN_l)_hIxI2_N

for

x

RN

(27rN/2/r(N/2)) is the volume of the unit sphere in a locally integrable function which satisfies where WN_1 =

T is

in the sense of distribution theory (see Theorem 3 in Section 5.4). Now the Laplacian operator can be extended to differential forms by defining

f

= The same formula (for I E

extends the Laplacian operator to currents.

LEMMA I

For a current FE

= d(dF) + d*(dF) where d is the £2-adjoint of d. PROOF

l'his is a straightforward calculation using the formula for d

d{fdx'}=

Adxt

j=I and

the formula for d* (see Lemma 2 in Section 1.5)

d{fdx'} I

Details will be left to the reader. We apply

to

the current T(x)dx where dx =

= t50dx =

On the other hand, d{Tdx} =

0

dx1

A ...

A

dxN. We obtain

[0].

because Tdx is a current of top degree.

Therefore, Lemma 1 yields

=

=

Exterior DerMxtlve and Cauchy-Rlemann Operaton

280

Letting k =

we obtain

dk = (i)N[O) The resulting fundamental solution for d is

K=r'k where r(y, x) = x — y for x, y E RN This kernel is called the spherical kernel. Using the formula for d, we can write down an explicit formula for k and hence K. We have k(x) =

A...AdXN}

= (_l)N

WN_1

where

A... A dXN)

j=I

lxi

indicates that ds3 has been removed. The last equality uses the

equation

8T(x) OX,

I

— WN_1 IxIN

0. Since x,lxJ_N easy safety disc argument or something equivalent shows that this equation holds in the sense of distribution theory across the origin as well. The corresponding spherical kernel is given by

which follows from a straightforward calculation for x is

a locally integrable function of x E

an

K(y,x) = (rk)(y,x) =

(1)

- Yt) A... A

A...Ad(XN—YN). Note that K has locally integrable coefficients on pN x RN, because

K(y,x)I $ Cjx — for some uniform constant C. It is an interesting fact (which will not be used in the sequel) that the spherical

kernel is equal to the average of the ray kernels measure dc on the unit sphere { loi = } in RN. 1

with respect to surface

Fundamenial solutions for a on C"

19.2

281

Fundamental solutions for

on C"

We start with the construction of the Cauchy kernel on a slice which is analogous to the ray kernel on Then we construct the Bochner—Martinelli kernel [Boc] which is analogous to the spherical kernel on RN. Cauchy kernel on a slice

To construct the Cauchy kernel on a slice, we start with the fundamental solution

for 0/02 on C given by

T(z)=-'--. As shown in Section 22.4, we have OT

C.

on

Oz

We give C" the coordinates (z,, z') where z1 E C and z' = (z2,. . .,

E

C"'. Define [0'] = i50(z')dv'

where

= (2i)'"d22 A dz2 A... A

A

is the volume form on C"'. c1(z,) and define c = c1 ® [0').

Since T is locally integrable on C, c is a well-defined current in V""''(C"). Now 0[0'] = 0 because [0'] is a current of top degree on C"-'. We obtain

(_T)d2I

Oc= = =

Adz, ®[0']

A dz1 ® [0'] 15o(zu)

=10].

0 i5r,(z')dxi

A dy1 A dv'

Exteijor Derivative and Cauchy-Rlemann Opeiutors

282

We define the Cauchy kernel on a slice by

C = r•c

r: C" x C" —, C" is given by r((, z) =z — (. Since r is holomorphic, r and 0 commute. Pulling back the equation Oc [0] yields where

= r'[O] =[Li]

on C"xC".

Therefore, the Cauchy kernel on a slice is a fundamental solution for 0. We summarize this discussion in the following theorem. THEOREM 1

The Cauchy kernel on a slice defined by

=

±d((i_zi)

is a biregular fundamental solution for 0. Here, i.e.,

®

is the diagonal in C"' x

= {(z', z'); z' E C"' }.

The Cauchy kernel on a slice is constructed as the tensor product of a funwith the diagonal in the other variables. We could damental solution for have used the fundamental solution to the Cauchy—Riemann operator along the

complex line in C" corresponding to an arbitrary point a in projective space,

CP"'.

In this way, we can construct a family of Cauchy kernels,

indexed

by a

CP"', and each one is a fundamental solution for 0. The collection a is analogous to the collection of ray kernels of kernels discussed in the previous section. Note that for the Cauchy kernel on the z1 -slice defined above,

supp C C

= {(' = z'}.

This support property will be crucial in the proof of Bochner's global CR extension theorem presented in the next section. Since C is supported on such a "thin" set, clearly the coefficients of C are not locally integrable on C" x C". A locally integrable fundamental solution for 0 is given by the Bochner—Martinelli kernel, which is our next topic.

Fundamental solutions for 0 on

283

The Bochner—Martinelli kernel As with the construction of the spherical kernel, we start with the fundamental solution for on C" given by

T(z) =

—(n

—

2)!

for

C"

(see Theorem 3 in Section 5.4). The analogue of Lemma 1 in the previous section for the Laplacian on C" is the following lemma. LEMMA

I

For a current F E VI* (C")

=4(aa*F+o*OF) where PROOF

is the £2 -adjoins' of 8.

The proof of this lemma is a straightforward calculation using the

formula for li A

=

A dzt A

31 and

the formula for ô*

8*{fdziAdEJ}

(see

Lemma 6 in Section 3.3). The factor of 4 comes from the fact that the

Laplacian operator on functions is given by Ti

Details will be left to the reader.

I

Let dv be the volume form for C", i.e.,

dv = dx1A dy1 A. .. A

Exterior Derivatiwe and Cauchy—RiemannOperators

284

where z3 =

j

+ iy3 for 1

n. By applying

A{Tdv} =

to T(z)dv, we obtain

i50dv

=[O].

Since dv is a form of top degree, clearly Ô{Tdv} =

0.

Lemma I yields

_488*{Tdv} =

=

60dv

=[OJ. We let

b= Using the previous equation, we obtain [0].

Define the Bochner—Martinelli kernel by

B=rb where_ T((, z)

=

z —

(for (,z E

Since T is holomorphic,

commutes

with t9. Therefore, we have

=

T*Ob

= r*[O1

So

the Bochner—Martinelli kernel is a fundamental solution for 8.

Our next goal is to obtain a working formula for the Bochner—Martinelli kernel which will be useful in later chapters. To do this, we compute b. By using the formula for we obtain b=

-

'

= —(ii--

A dz1 A ... A

A dz1 A ... A dij A

A.

. .

A

A

0. Since last equation follows from a straightforward calculation if z this equation also holds in the sense is locally integrable in z E of distribution theory across the origin.

The

Pundizeatal solutions for 0 on

285

Define

=

j=I dz

=

dij A

We obtain b= Pulling this back via r' yields our desired formula for the Bochner—Martinelli

kernel. We summarize the above discussion in the following theorem. THEOREM 2 The Bochner—Martinelli kernel

B((, z) =

((( — z) d(( — z))A(d((— z)

is a biregular fundamental solution for

on

—

C".

The Bochner—Martinelli kernel is analogous to the spherical kernel on RN. This analogy can be carned one step further. It can be shown that the Bochner— Martinelli kernel is the average of the Cauchy kernel slices C0 with respect to (see [HP]). the Fubini—Study volume form on If n = 1, then

B((,z) = So in one complex variable, the Bochner—Martinelli kernel reduces to the Cauchy kernel. Suppose f is a smooth function defined on a simple closed contour in the complex plane. Define

F(z) = B(['y]°"f)(z) —

—1

f

2iriJ (—z It is a classical fact that the boundary value jump of F across 'y (from outside to inside) is precisely f. It is our goal to generalize this jump formula for the Bochner—Martinelli kernel. l'his jump formula will be used in the proof of Bochner's global CR extension theorem given in the next section. It also will be of crucial importance in the construction of the solution to the tangential Cauchy—Riemann equations in later chapters.

Exterior Derivative and Cauchy-Rl.mann Operators

286

Let M be the smooth boundary of an open set D in C". Let I E D*(M). Since B is biregular, B([M]°" A f) is a well-defined current on C". In fact, B([M]°" Af) is a current on C" with locally since IB((, z)I CI( — integrable coefficients. In addition, B((, z) is smooth for z. Therefore, B([M]°" A f) is a smooth form on C" — M. Our intention is to show that I is the boundary value jump of B((M1°" Al) across M. If M is noncompact, then we must require the compactness of supp f. If M is compact, then D*(M) = (*(M) and so f can be any smooth form on M. To state this boundary value result, we need some additional notation. For a form f 6p,q (D), we say that F has continuous nontangential boundary values on M = OD from D if for each zO M Z

lim F(z) Z()

exists

in D

is any nontangential cone in D with vertex at zO. A cone where is nontangential if there exists a A > 0 such that

K—zIAK—zol

for

We also remind the reader that for I E the tangential piece of AP.QT*(M). Suppose D = {z E C"; p(z) <0} where I M is denoted ftM IR is smooth, with dpi 1 on M. Let N = p: C" be the dual vectoT to ap. As shown in Lemma 2 in Section 8.1, ftM

and f are defined on a neighborhood of M in C". So ftM defined on a neighborhood of M in C". Now N,

is

also

THEOREM 3

Suppose M is the smooth boundary of a domain D in C". Assume M has the induced boundary orientation as the boundary of D. Let D D and =C" —D. Suppose wit/iC', compactly supported coefficients. Then

{B([MJ°" Af)}tM ID+

and {B([M]°" Af)}tMID_

have continuous nontangential boundary values on M from D+ and D, respectively, denoted by B+f and Bf. Moreover, on

M.

More delicate boundary value results are mentioned in Chapter 24.

PROOF The idea is to reduce the proof of this theorem to the case where f is a function, where the proof is easy. We need the following lemma (from [I{PJ).

Fundamental solutions for 0 on

287

LEMMA 2

There is a constant C > 0 such that for any multiindices 1 and J B(çT, z) A

A

= B((, z) A

A

+A1

A dz'

A

z) +

A

A and A2 are differential forms which are smooth for ( there is a constant C > 0 such that

<

for (,z E

z. Moreover,

—

in some neighborhood of M.

Assuming Lemma 2 for the moment, we complete the proof of Theorem 3. Let

f(() =

d(' Ill = p hI = q

A

whose coefficient function Ii is C' with be an arbitrary (p, q)-form on compact support. If Al = {p = 0} where IdpI = 1 on M then

where p denotes Hausdorif (2n — 1)-dimensional measure on M (see the end of Section 6.1). From Lemma 2, we have

B([M)°" A f)(z) =

z),

A

=

A

A dz1 z), A

The tangential piece of the third term on the right vanishes due to the presence of Op. We obtain

{B([M]°" A f)(z)}LM =

A dz'}IM

+

f

A1 ((, z) is the coefficient of the piece of A, ((, z) of bidegree (n, n) in (. Now let zO E Al and suppose is a nontangential cone in either D - or we have D+. For some A > 0 depending only on the aperture of

K—zIAK—zoh

for

Exterior Derivative and Cauchy-Riemann Operators

288

Combining this with the estimate on A1 given in Lemma 2, we obtain

Cj( —

sup

for

zEC2

(E M.

is integrable in ( E M, the dominated convergence theorem

Since K —

implies that

J

CE M

exists and that this limit is the same regardless of whether or not in or D. So the boundary value jump of

is contained across

Al at zo vanishes. has continuous nontangential Therefore, it suffices to show that B([M}°'1 boundary values from D and D+ and that the boundary value jump across M In other words, we have reduced the proof of Theorem 3 from D+ to D is to the case where f = is a C' function with compact support. Since the Bochner—Martinelli kernel has only diagonal singularities, B([M]°"f,) is smooth (C°°) on — {supp fi}. In particular, the boundary value jump across M — {supp 11 } is zero. Thus, it suffices to prove Theorem 3 at points in {supp ft fl M}. Fix E {supp fi}flM. Let E with I on a neighborhood of

{supp f'}. Define

9i(() = =

—

Note that g, + h1 = 1'. Applying part (b) of Theorem 2 in Section 18.2 to the degree zero current T = xDgI, we obtain

= —B([M]°"gi) + (Note that B(T) 0 since T has degree zero). Since = 0 near zo, the second term on the right is smooth in a neighborhood of z0 in C' and so its boundary value jump across M is zero near From the above equation, B([M]°"gj) has a smooth extension to M from D = D and D+ = C" — D and the boundary D is value jump of B([M]°"g,) across M from = fi(z.o), near z0, i.e.,

B(g,) = fi(zo) on M

near z0.

Since

is of class C', we have Ih,(<)I

— zol

(1)

Fundamental solutions for Ô on

289

for some uniform constant C> 0. As with A1 above, this estimate yields

((E M)

sup IB(C, ZEC20

where

is any nontangential cone contained in either D+ or D. By the

dominated convergence theorem z€czo

and this limit is the same regardless of whether or D+. Therefore, its boundary value jump across M at exists

—

is contained in D vanishes, i.e.,

B(hj)(zo) = 0.

(2)

Since ft = + h1, we see that B([M]°'f1) has continuous nontangential boundary values on M from D and Equations (1) and (2) imply

B(fi)(zo) = fi(zo) as

desired. This establishes Theorem 3 for functions ft. and so the proof of

Theorem 3 is complete. PROOF OF LEMMA 2

I

Let N be the dual vector to Op. In particular

= Let

z) = B((, z) A From the product rule for j

(see

A

A

d('

—

A dz').

Lemma 1 in Section 1.5), we have

=

Ai((,z)

A D((,z))

A2((,z) = The term A A2((, z) gives the third term on the right side of the equation stated in Lemma 2. So it suffices to show A1 z) satisfies the estimate stated in the lemma. Actually, we shall show A

z)I CK —

for a uniform constant C.

From the expression for B((, z) given in Theorem 2, we may write Ad(( — z)

(3)

290

where

Exterior Derivative and Cauchy—Riemann Operators

a is a form of bidegree (0, n —

Since

— z) A (d(3 — dz3)

1)

in (.,z and where

= 0, we have

B((, z) A d(' = B((, z) A dz'. From the definition of D, we therefore have z) = B((, z) A Ôp(() A

=

Since Op(() A

t9p(z) A D((, z)

0,

—

d2') A dz1.

we obtain

=

A B((, z)

—

A

—

—

d23

A dz1. (4)

We have

9p(z)

=

—

(5)

31 Since

IB((.z)I <

we have

—

B((,z)I <

—

(6)

—

for some uniform constant C. We may also write

where

a

is

a form of bidegree (n, 0) in ((.z) and where d(( — z)

d(( — z)

A

= d((1

—

z1) A

... A

= 0, we have

—

d(23

—

(,) AB((,z) A

=0.

This together with (4), (5), and (6) yields the estimate stated in (3). From the

definition of A1, the estimate in (3) implies IA1

((, z)I CI( - z12_2n

for z E near M where C is a uniform constant. The proof of Lemma 2 is now complete. I

Rochner's global CR extension theorem

291

Bochner's global CR extension theorem

19.3

The Cauchy kernel on a slice and the Bochner—Martinelli kernels can be used to give an easy proof of Bochner's global CR extension theorem, which roughly states that any CR function of class C' on the smooth boundary of a bounded

domain D in (n > 2) extends to a holomorphic function on D. We first recall that both the Bochner—Martinelli kernel (B) and the Cauchy kernel on a slice (C) are biregular kernels. Therefore, we can apply B and C to currents with compact support (Lemma 2 in Section 18.1). Ultimately, the current we have in mind is T = [MJ°" f where M is the smooth boundary of the domain D and f is our given CR function on M. This current is 0-closed since f is CR (Lemma 5 in Section 8.2). As the next lemma shows, the Bochner— Martinelli and Cauchy kernel on a slice both agree when applied to 8-closed compactly supported currents of bidegree (0, 1). This together with the support property of the Cauchy kernel and the jump formula for the Bochner—Martinelli kernel will yield the proof of Bochner's theorem. LEMMA Suppose

(n

1

-

T is a 0-closed, compactly supported current of bide gree (0,1) in 2).

Then

(a)

C(T)

0 on the unbounded component of

(b)

C(T)

B(T) as currents on

PROOF

—

supp T.

Give C'2 the coordinates (zi,z') with z1 E C and z'

E

C'2'. From

the formula

d((((;IzI))

C((, z) = we see that supp C c {(' = z'}.

Therefore,

supp T C

® K' = z']

if

{((i,('); WI
then

supp C(T)

C {(z,,z'); Iz'I

< R}.

Intuitively, this is because if z' is fixed with Iz'I R then the c-support of C((, z) msses supp T. It is here that we have used the assumption n 2. Since OT = 0, part (b) of Theorem 2 in Section 18.2 yields = T. C(T) is a holomorphic function on C'2 — supp T. Since C(T) vanishes for {Jz'I R}, part (a) follows from the identity theorem for holomorphic So

functions.

Exterior Derivative and Cauchy-Riemann Operators

292

For part (b), first note that since both B and C are fundamental solutions for 0 and since 07' 0, we have

l){B(T)-C(T)}=T-T=o on Therefore, B(T) — C(T) is an entire function on C'2. We claim B(T) — C(T) vanishes at 00 and therefore it vanishes identically by Liouivile's theorem for entire functions. From part (a), we know C(T)(z) = 0 for Izi large. So it suffices to show B(T) vanishes at 00. Let Il be a neighborhood of supp T. For fixed z B((, z) the form we have is smooth for ( E ft For z

B(T)(z) = = where, as before, the notation (, indicates that (is the variable of "integration." Since T is a continuous linear functional on there must exist a constant C> 0 and an integer N > 0 such that

C

sup

IaIN

CE supp T

for

z

converges

By

examining the formula for B((,

to zero as z —' 00. Thus,

z), we

see that the right side

B(T) (z) —' 0 as z)

—' oo,

as desired.

I Part (b) of the lemma is useful since the Cauchy kernel and the Bochner— Martinelli kernel each has properties not a priori possessed by the other. The Cauchy kernel has a nice support property described in part (a) which is not apparent for the Bochner—Martinelli kernel. On the other hand, the Bochner— Martinelli kernel has nice regularity properties since B((, z) has locally in-

tegrable coefficients. Part (b) states that if T is a 0-closed compactly sup-

ported (0,1)-current, then C(T) and B(T) both enjoy these properties since B(T) = C(T). We now state and prove Bochner's global CR extension theorem. THEOREM I

Suppose D is a bounded open set in C'2 (n 2) with smooth boundary. Suppose f is a CR function on OD of class C'. Then there is a holomorphic function F on D which has a continuous nontangential extension to OD from D such that FIOD = f. Moreover, F is given by either of the following integral formulas:

F = —B([OD]°"f) = —C([OD]°"f).

Bochner's global CR extension theorem

293

Note there are no convexity assumptions on OD for this global theorem.

Let M = OD. 1ff is aCR on M, then from Lemma 5 in Section 8.1, (MJ°" f is a 8-closed, compactly supported current of bidegree (0,1) on C's.

PROOF

Let

F = —B([M]°"f). By part (b) of Lemma 1, we have

F= From Theorem 3 in Section 19.2, From part (a) of Lemma 1, F 0 on nontangential extension to M from D. In addition, the F has a continuous — D) is equal to f. Since boundaxy value jump of F across M (from D to

F is holomorphic on D. Since

0,

Theorem 2 in Section 18.2 yields

OF = —[M]°"f which has support in M. Therefore, F is holomorphic on particular on D), as desired.

—

M (and in

I

Note that Bochner's theorem does not hold for domains in C'. This is because every function on a closed contour in C is a CR function (there are no tangential Cauchy—Riemann equations). The above proof breaks down in n = 1 because part (a) of Lemma I does not hold for n 1. For a simple closed contour in C, the condition that replaces 8M = 0 for Bochner's theorem is the moment condition, which means

f

J

for

n=0,l,2

Note by Cauchy's theorem, this condition is necessary for f to be the boundary values of a holomorphic function defined on the inside of y. To see that this is a sufficient condition, let

F(z) = —

1

[

2rriJ (—z <E.y

Theorem 3 in Section 19.2 still holds (regardless of whether or not f satisfies

the moment condition). In particular, the boundary value jump of F across 'y (from the inside to the outside of y) is equal to f. From a series expansion of 1/(( — z) in powers of (together with the moment condition, we see that F(z) = 0 for z outside 'y. It follows that the boundary values of F from the inside of -y agree with f.

20 The Kernels of Henkin

In Chapter 19, we constructed two fundamental solutions for 0 on

As

mentioned in Chapter 18, if K is a fundamental solution for 0 and if f

with Of

= K(f). with Of

0, then the equation Ou =

f

can be solved on

by setting Now suppose D is a bounded domain in CT' and suppose f E 0 on D. In this case, we cannot directly apply K to I without first

extending f and then multiplying by suitable cutoff function so that f has compact support. This process produces an extended f which is no longer 0closed. Another way to cut off f is to multiply f by the characteristic function on D (denoted If Of = 0 on D, then

= —[OD}°'

A 1.

As mentioned in Section 18.2 (see Theorem 2), we have

9{K(xDf)}

—

A 1) =

Therefore, the term A f) is the obstruction to solving the equation Ou = on D. In this chapter, we define a general class of kernels due to Henkin which allows us to solve the equation thj1 = K([OD]°" Af) on a strictly convex domain D and so in this case, u = K(XDI) — m solves the equation Ou = on D. We should also mention that a slightly different kernel approach to the solution of the equation Ou = f was discovered by Rarnirez IRa].

f

f

20.1

A general class of kernels

The kernels we are about to define are due to Henkin; however we shall employ the more streamlined notation of Harvey and Policing [HP].

294

A general class of kernels

Let V u3: V

be

295

1<j

x Ctm. For each an open subset of is a smooth map. We write

< N, suppose

u3((,z) = and

we use the notation

. (ç -

z)

- zk)

=

.d((-z)

.d((

-zk)

- z) =

Ad((k - zk).

Here, Ô refers to the Cauchy—Riemann operator on

x

(i.e., in both (

and

z).

For 1 <j
is smooth on the set V — A3

{(ç,z) E

where

A3

V; u3((,z) ((—z) =O}.

Let N be the Set of nonnegative integers. Let

For a E set + For an increasing multindex J {i1,.

For

aE

.

with I

.

i3

N, let

let

= Here,

a is the Cauchy—Riemann operator on

x By definition, the form the wedge product of with itself times. Note that since a 2-form, is typically not zero. With this notation, we define the is

(9w,

kernel

=

A

=fl —p

The Kernels of Henkin

296

We also define E1 =0 if I

that E1 = 0 if I

Note

N" with p > n. The

Sometimes for emphasis, form E1 has smooth coefficients on V — we shall write E(u",. . , for E1. The following lemma yields simpler formulas for the E1. .

LEMMA 1

For k

0 A

for ((,z)

PROOF

(w,)

V—A,. From the formula d(( — z)

•

•

((— z)

we have

'

—

—

u'.((—z)

The wedge product of w, with the second term on the right vanishes due to the repeated wedge product of the 1-form u' d(( — z). The lemma now follows. I Using this lemma, we single out some special cases of interest. With p = 1.

we have

= E(u') =

A

For example, from the expression for the Bochner—Martinelli kernel given in

Theorem 2 of Section 19.2, we have

B=E(u) where u((, z) = (— z. With p = 2, we have

E(u',u2) =

A

A •

(

— z) 1

((_ z) I

A formal identity

20.2

297

A formal identity

We now prove a formal identity which relates the above-defined kernels to the 0-operator. Suppose I = {i1,... , i,} is an increasing muitlindex. For 1 j p. let

The notation i3 means that i, has been omitted. So

is an increasing index of

length p—i. THEOREM I

Suppose I = {i1,. .

.,

is an increasing index of length p.

8E1 = on V — (b) Suppose that M is a CR submantt'old of C'1, then OM MEl = on {M x M} n {V — (a)

The following special cases are important. If p = 1, then

on V—A1. This generalizes a result we already know for the Bochner—Martinelli kernel z}). (since OB [is], clearly OB 0 on {((,z);

If p= 2, then

on V—{A1UA2}. If p= 3, then E12

The analogous identities hold for PROOF

on V —

—

{A1 U

uA3}.

M.

From the definition of E1 and the product rule for 0, we have

jI jI

IaI=n—p

w1'A )

jaI=n—p+I

The Kernels of Henkin

298

If = 0, then the expression within the brackets is exactly E13. Therefore, we have

=

+

j1

A

31

IaI=n—p+1

proof of part (a) will be complete provided we show the second term on the right vanishes. In fact, we will show The

=0

A

foreachmultiindex (1), we use the vector field

(1)

Toestablish

From the formula for w3, we have

From the product rule for j, we obtain -

—%'•((—z) &

+

(u' .((_z))2

=0. We also note WI A

=0

for = n — p + 1 because the form on the left involves a wedge product of degree n + 1 generated by {d((1 — z1),.. . , — and therefore one of the d((, — z,) must be repeated. Using this together with Ojw3 = I and 9j 8w3 = 0, we obtain 0 = OJ(W' A (9WI)0r)

A

This proves (1) and thus completes the proof of part (a). Part (b) follows by

taking the tangential piece of the equation in part (a) and by using the definition of the (extrinsic) 8M M complex (see Section 8.1). I

The solution to the Cauchy—Riemann equations on a convex domain

299

203 The solution to the Cauchy—Riemann equations on a convex domain For a given convex domain D, we shall define an appropriate map u which when inserted into the kernel machinery of the last section will yield a solution to the 8-problem. The resulting kernels will also be used in subsequent chapters to solve the tangential Cauchy—Riemann equations on a strictly convex hypersurface. Although our focus will be on a strictly convex boundary, we will indicate in Chapter 24 how the kernels can be modified for strictly pseudoconvex and other geometries. Our eventual goal in Chapter 22 is the local solution for the tangential Cauchy—Riemann equations on a strictly pseudoconvex hypersurface. From Theorem 1 in Section 10.3. a strictly pseudoconvex hypersurface is locally biholomorphically equivalent to a strictly convex hypersurface. In addition, solvabilty of the tangential Cauchy—Riemann complex is invariant under a biholomorphic (or more generally, a CR diffeomorphic) change of variables

(see Corollary 2 in Section 9.2). Therefore, the kernels that locally solve the tangential Cauchy—Riemann equations on a strictly convex hypersurface will also provide a local solution to the tangential Cauchy—Riemann equations on a strictly pseudoconvex hypersurface. Let us suppose that D is a strictly convex domain in PJ. Let p be a smooth defining function for D, i.e.,

D =

with smooth boundary

<0}.

E

Since D is strictly convex, the real hessian of p at a point on M is positive definite when restricted to the real tangent space of M. By replacing p with

p+ Cp2, for a suitable positive constant C, we may assume that the real hessian of p at a point on M is positive definite in all directions in including the normal direction to Al. This allows us to prove the following lemma. LEMMA 1 Let D = {z e compact set in

(a) (b)

p(z)

<0} be a strictly convex domain

and suppose K is a

There exists e > 0 such that if (, z E K with — There exist constants f >0 and 0< C1

and

—

<€, then

-p(z)

.(( - z)} <

The kernels of Henkin

300

Tç(M,)

I p
(p>r)

M,

FIGURE 20.1

Since p is a strictly convex defining function for D, there exists such that for each t with < t < e the level set PROOF

>0

= {( E K C C";p(() = t} is strictly convex. If ( E then {w E C"; be identified with the holomorphic tangent space, (w1,.

Therefore

'—b

. .

=

.

w = 0}

can

via the map w =

if p(() =

t

and if (ôp(()/O()

then the vector z — ( belongs to The strict C convexity of which completes implies that if ( z then p(z) > t = the proof of part (a). For part (b), we first note that for some 0 the real hessian of p at (is positive definite provided —c < p(() < c. Therefore, part (b) follows from a second-order Taylor expansion of p about the point (. I ((— z)

0

C" x C"

Now we define the functions

C" for j = 1, 2, 3, which will

generate our desired kernels. Let

(ôp(()

Op(()

— Op(z) — (Op(z)

ap(z)

—

u'((,z) — —

u2((,z)

O(

—

u3((,z) = (— z.

—

—

*9z1

The solution to the Cauchy-Riemann equations on a convex domain

301

Note that u1 only depends on ( and u2 only depends on z, but we wish to think of both u1 andu2 as being defined on Ctm x

Using (L1,U2,U3, we define the kernels E1 E(u'), E2 = E(u2), E3 = E(u1, u3), E23 = E(u2, u3), and E123 = E(u3), E12 = E(&, u2), E13 u2, u3) as in Section 20.1. For historical reasons, we assign the following labels to these kernels:

L=

E1

(after Leray)

H=E13 (afterHenkin). As already mentioned, the Bochner—Martinelli kernel is given by

B=E3. By definition, the transpose of a kernel is obtained by switching ( with z. Since u3(z,() = —u3((,z) and u2(z,() u'((,z), we obtain

Ht((,z)

= E23((,z)

Lt((,z)

= E2((,z)

and

Bt=B. Recall that up to a sign, the transpose of a kernel is equal to its adjoint as an operator on forms (see Section 18.1). The L and H kernels are smoothly defined on the set

{

((,z)

E

x Ctm;O

((-z)

In view of part (a) of Lemma 1, this set includes the set {((,z) E M x D}. Reversing the roles of ( and z, we see that Lt and Ht are smoothly defined on the set {((, z); ( E M and 0 < p(z) < }. From Theorem I in Section 20.2, we have

OH=L-B onMxD. on{((,z);(EMandO
Suppose K is a compact set in Ctm. For f E V*(K) the forms L([M]°" Al) A 1) and H([M]°" A f) are smoothly defined on D and the forms Af) are smoothly defined on the set {z E < p(z) < c}. and

302

The

Moreover, there is a constant C > 0 (independent

C

IH([ODJ°" A

sup

Kernels of Henkin

of f E D (K)) such that sup

zEDflK PROOF Since the forms

H and L have smooth coefficients on the set {((, z) E M x D},

H(fM]°" A f)(z)

A

=

f)(z) =

J(EM [H((, z) A - J(EM

[L((,z) A Af) and Lt([M]o.t A

depend smoothly on z E D. Similarly, the forms

We shall defer the proof of the estimate given in the lemma until the next section where we shall prove a more general result. I Another key fact is that L and Lt act nontrivially only on forms of highest

and lowest degree, respectively, as the next lemma shows. LEMMA 3

Suppose f E

Then

L([M]°" A

A f) = 0 unless q = n — 1.

f)

0 unless

q

0.

In addition,

depends only on (, and examination of the formula for L((,z) shows that the degree of L in d( is n — 1. Therefore, PROOF

Since u1

z) =

L([M]°" A f)(z) =

f has degree 0 in

-J

IL(C, z) A

—

Similarly, the degree of Lt((, z) in d( is 0 and so Lt([M}°" A 1) = 0 unless the degree of f in d( is n — I. I

Now we present Henkin's integral kernel solution to the Cauchy—Riemann

equations on a strictly convex domain. His solution has an L°° estimate which was one of the motivations for the construction of integral kernel solutions to the Cauchy—Riemann equations.

Boundary value results for Henkin 's kernels

303

THEOREM 1

(See fHe2].) Suppose D is abounded strictly convex domain in

boundary M. Let f E

q

I

n with t9f =

0

with smooth

on D. Then the ftrm

+ H(jMj°' A f)

=

f

is a solution to the equation = on D. Moreover, there is a constant C > 0 which is independent of f such that D

D

We apply part (b) of Theorem 2 in Section 18.2 to the fundamental and since 9f = 0 on D, we solution B with T = xof. Since PROOF

obtain

ô{B(XDf)} — Using the equation OH = L A

—

A

A f) ± L([M1°' A

and

(1)

B on M x D, we obtain

f)

Since f E

1) = XDf

f)

on D.

q I, the second term on the right vanishes by

Lemma 2. Using the homotopy formula in part (b) of Theorem I in Section 18.2, the first term on the right equals —O{H([M}°'

Since a{fM}°' A 1)

f) + H(O{[MJ°' A f})}.

0, we obtain

B([M1°'

A

f) =

—O{H([M]°1

A

f)}

on D.

Inserting this into (1) yields our solution n. The estimate on Iul follows from the estimate given in Lemma 2 and from the fact that the Bochner—Martinelli kernel B((, z) is uniformly integrable in

(EDforzED.

20.4

I

Boundary value results for Henkin's kernels

In this section, we shall prove the estimate given in Lemma 2 of the previous section. We shall also examine the smoothness of these kernel operators up to the boundary of our given convex domain. The boundary values of these kernel operators are the key ingredients for the construction of one of the fundamental solutions for the tangential Cauchy—Riemann complex on the boundary. Let D = {z E C'; p(z) < 0}, let = {z E C?z; p(z) > 0}, and let

M be the boundary of D. From Lemma 2 in the previous section, the forms

304

The Kernels of Henkin

A f) and L([M}°' A f) are smooth on D and A f) and Lt([M]oi A f) axe smooth on the set fl U where U is a neighborhood of M in We shall examine the regularity of the boundary values (on M from D) of H([M}°" Af) and Af) along with their tangential derivatives. Likewise, we shall examine the regularity of the boundary values (on M from of Ht([M}°"Af) and Lt([MJ°"Af) along with their tangential derivatives. Recall that a vector field X on is considered tangential to M if Xp = 0 near M where p is the defining function for M. Our main boundary value results for Henkin's kernels are contained in the following two theorems.

THEOREM I

Suppose D is a strictly convex domain with smooth boundary M. There is a neighborhood U of M such that the following holds. Let N be any nonnegative integer and suppose X1, .. , are tangential vector fields to M. If f is a compactly supported (p, q)-form with coefficients of class CN on lvi, then .

X1

. .

.

X.v{H(EM)°' A f))ID- and X,

A

. . .

continuous extensions to M. Moreover, for any compact set K in a constant C > 0 which is independent of f such that X1

D-nK sup

have

there is

.. XN{H([M]°" A f)}I

X1

.

.

. .

A f)}I

{ D+ riUnK }

for any smooth form f on M with support in K. Here, If ICN(M) is the usual

of f on Zt'i.

THEOREM 2

Suppose D is a strictly convex domain with smooth boundary M. There is a neighborhood U of M such that the following holds. If f is any compactly supported (p, q)-form with coefficients that are of class CN+I on M, then any derivative of order N of have A f)ID- and of V([M)°' A continuous extensions to M.

If D is bounded, then the above two theorems also apply to forms without compact support. We remark that the case N = 0 (i.e., no derivatives) is given in [He2] (see also [He3]). Some refinements of these boundary value results due to Harvey and Polking are given in Chapter 24.

In Theorem 2, we are allowed to take normal derivatives of L([M]°t A f) and Lt([M1oI A f), whereas in Theorem 1, we are only allowed to take tangential derivatives of H(1M]°' A f) and Ht([jll}o.l A f). There is no loss of differentiability in Theorem 1. That is, the form f is assumed to be of class CN and we obtain a This contrasts with Theorem 2 where there

Boundaiy value results for Henkin 's kernels

305

is a loss of differentiability. As we shall see from the proofs, this is due to the

fact that L((, z) and Lt((, z) are not integrable in ( E M for fixed z E M, whereas both H(C, z) and Ht((, z) are integrable in E M for fixed z M. The rest of this section is devoted to the proofs of these theorems. The basic idea of the proofs is to localize and then make a change of variables that flattens out it'!. The strict convexity allows us to estimate the resulting kernels. We then show that tangential derivatives do not worsen these estimates.

We shall prove these theorems for H and L. The proofs for Ht and Lt similar. By examining the L and H kernels, we see that we must analyze a term of the form

K(f)(z)

=

J(EM K((, z)f(()

where k((,

K'\

p+q—n, p, q_>

where f is a function of class CN on M and where dc(() form on M. For II, k is a smooth function satisfying

denotes

the volume

z) = O(I( — z()

both p and q are at least 1. For L, k is simply a smooth function without any estimate and q = n, p = 0. We fix a point z0 E M and examine the regularity of K(f)(z) as z approaches M from D near zo. Fix (o E M. If (o z0 then by Lemma 1, ((Op(C)/O(). — z)) 0 for x z) in some neighborhood Ui x U2 C of ((h, zn). If 0 is a smooth compactly supported function in U1, then K(Øf) is smooth on U2. In particular, K(Of) is smooth up to M fl U2 from U2 fl D. By a partition of unity argument, it suffices to assume Co z0. That is, we may assume that f has compact support in a set of the form U fl M where U is an open neighborhood of Zo in (to be chosen later). We must show that K (f) has the desired regularity on U fl D. We need the change of variables given in the following lemma. and

LEMMA 1

For each z0 e M,

'I': U x U (a)

(b)

there is a neighborhood U of zo in

with

and a smooth map

the following properties:

(p(z),0,. ..,0). If we write W((,z) = (w1((,z),.. W(z,z) =

.

E

then

.(C_z)}.

306

(c)

The Kernels of Henkin

Foreach z EU, the map

= W((,z) is a

given by

: U '—+

from U to PROOF

We let

.((_z)} as required by (b). We have = dp(zo) + ilm{Op(zo)}

((,

= dp(zo)

—

ZJ*d()

where the last equation uses the fact that Op = 1/2(dp

—

iJdp)

(see Lemma 5

in Section 3.3). The real vectors dp(zo) and J*dp(zo) span (over R) the 1complex dimensional subspace generated by Op(z0). Since Op 0 on M, we can find vectors w2, .. . , so that {(Op(zo)/e9z), w2,.. . , form a basis for C'2 over C. We let

=

. .

where w1 ((, z) is defined above and

w3((,z) =

to3

((— z)

for 2 j n.

The real (-derivative of 'I'((, zo) at (=

zO is nonsingular. So property (c) follows from the inverse function theorem. Property (a) follows from the definition ofW. I

From

now on, we require f to have support in U fl M where U is a neigh-

borhood of

which is small enough to satisfy Lemma 1. We shall also require

U to be small enough so that the following estimate holds (from part (b) of Lemma 1 in Section 20.3) for(,zEU.

Re{wi((,z)} = p((), contains a neighborhood of the origin inthecopyofR2'2' given by {w C'2;Rew1 =0} for each fixed z MflU. After pulling back the integral in K(f) to this copy of R2'2' via 'I';', we Since

obtain

K(f)(z)

=

f

{Rewi =O}

Kj(w,z)fi(w,z)dv(w)

307

Roundwy value results for Henkin 's kernels

where

K1(w,z) = (p(w,

k1(w,z) —1

(1)

(w) — zI2P

dv(w) = volume form on {Rewi

0}

k1 (w, z)dv(w) = k('Içt (w), z)W' (dc)

fi(w,z) p(w,z) =

p

(w))

(w) — z).

Since f has compact support in U, there is an > 0 such that the w-support of fj(w, z) is contained in {w E C'1; wi
Yi

where Yi =

1mw1.

(2)

w') where Yi = 1mw1 e We shall give {Rewi 0} the coordinates w = and w' E Certain components of the original kernel such as k (for the kernel H) and . ((— z) and IC — z12 vanish for ( = z. These terms can be estimated by some power of (— z . We wish to transfer these estimates to the w-variables via the diffeomorphism (= 'Ic' (w). We shall use the following notation. For a smooth function g: C2 x C'2 '—' C and a nonnegative integer j, we say

g= provided g is a homogeneous polynomial of at least degree j in the real coor-

dinates of (p(z), yi, w') over the ring of smooth, complex-valued functions on C'2 x C2. A typical term of such a g is of the form

a smooth, complex-valued function. We say

g

= O(p(z),y,,w')3 and g(w, z) €j(p(z),y,,w')i3 for some uniform>0.

With the above notation, we state and prove the following estimates. LEMMA 2

(a)

Ifk((,z)=O(I(—zI')forjO,:hen k('I';'(w),z) = O(p(z),y,,w')3.

The Kernels of Henkin

308

(b)

Forw=(yi,w') jW'(w) —

(c)

I(p(z),

w')12

Forw=(yi,w') Re{p(w,z)} +

Parts (a) and (b) follow from a Taylor expansion of k('Iç'(w),z) — z in w about the point w = (p(z),0,. . . ,0) and the fact that or W:'(p(z),O,. ..,0) = z. Part (c) follows from the fact that W;'{Rew1 = 0} c I M = {p(() = O} and the estimate in part (b) of Lemma I in Section 20.3. PROOF

Now suppose X2 is a vector field involving z or

derivatives. In general,

= O(p(z),yi,w')'' (note the exponent decreases by one). This is clear from (3) because } = apa_I = 0. In this However, if X2 is a tangential vector field, then case, if f = O(p(z), , w')3 then also X,f O(p(z), , w')J. This is the key

if f = O(p(z),yi,w')' with j 1, then

fact which we will use to show that the estimates on our kernels do not worsen when we differentiate them with a tangential vector field. LEMMA 3 Suppose

(a)

is a tangential vector field to M.

Ifk((,z) =

O(K— ZIG)

thenfor w= (y1,w') z)} = O(p(z), yi'

(b)

Forw=(y1,w') = O(p(z),y1,w')2.

Part (a) follows immediately from the observations made preceding the statement of the lemma. For part (b). first note PROOF

p(w, z) = Re{p(w, z)} + lI/i. Since

=

=

=

0,

part (b) follows from part (c) of Lemma 2. I

COMPLETION OF ThE PROOF OF THEOREM 1

k((,z) =

As already mentioned, for H,

— zI) and so

ki(w,z) = O(p(z),yi,w').

Boundary value results for Henkin 's kernels

Together with the other estimates in Lemma 2, we obtain for w = C(Ip(z)I + Ii'iI + lw'I)

IK1 (w, z)I

Since p(z)

w')

D, we obtain

0 for z

IK1(w,z)I <

(Iw'l2 +

C is some uniform positive constant. X' are tangential vector fields to M. Differentiating Now suppose involves a sum of terms of the form Kf(z) with Xi,..

where

...Xf'{f1(w,z)}) with 0 m N. The term involving the derivatives of in absolute value by If ICN. We now show that the same estimate that is satisfied by IK1(w, z)I.

is dominated above satisfies

LEMMA 4

are tangential vector fields to M. There is a

Suppose

positive constant C such that ..

for w = (Yi, w') and z PROOF

z)}I

.

+

D.

This lemma follows easily from Lemma 3. For example, we have —q

}——q

X2p(w, z) (p(w.

which by Lemmas 2 and 3 is dominated above in absolute value by C(Ip(z)2I + ly' + Iw112) + p(z)2 + 2 + Iw'I2 + fyj

Since p(z) 0 for z E D, this term is dominated by C (lwhI2 +

which is the same term that dominates I(p(w,

Thus, differentiating p(w, z) —Q with a tangential vector field does not worsen the estimate. I Now we show that the function

(Jw'J2 +

(p + q = n), q 1

The Kernels of Henkin

310

z)}I for z E D is For once this is done, the dominated convergence theorem will allow us to take limits of Kf(z) or as approaches ]t'f from D. This will complete the proof of Theorem 1.

which dominates both IK1(w, z)I and .. a locally integrable function of w = (Yi, w') E . .

.

LEMMA 5 The function

g(w)

q 1

p + q n,

= (lw'?2 +

is a locally integrable function of w = (Yi, w') E

First suppose q> 1. By integrating

PROOF

[ J

[

we obtain

dyidv(w')

[

(lw'?2 + Iyi l)QIw'l2P'

J

—

Iw'I
dv(w')

J

where C is a uniform positive constant. Since this last expression is an integral

in w' E

it is easily shown to be finite by a standard polar coordinate calculation. If q = 1 then the above integral in y' results in a log term but otherwise the calculation is no different than the case q> 1.

PROOF OF THEOREM 2

For the L kernel, we must analyze a term of the form

Ki(w,z) = z

U

k(w,z) (p(w,

D-, where k is a smooth function. From part (c) of Lemma 2 C + lyi

1K1 (w, z)I

for p(z) < 0. The right side is not integrable in w = (y1,w') E Therefore, an integration by parts argument must be used to reduce the exponent in the expression for K1. This explains the loss of one derivative in the statement

of Theorem 2. Recall that p(w, z) = Re{p(w, z)} + iyi. We have Op(w,z)

= —Rep(w,z)+z. 81/i

OYi

In particular,

)p(w, z)

Since n 2, we have

= ((1

0.

Boundary value resulls for Henkin's kernels

Since fi(w,z) = f(W;'(w)) with

has

311

compact w-support, we may integrate by parts

and obtain

f f(p(w,

Kf(z)

(w, z)k(w, z)dyidv(w')

ii

=

dyidv(w'). From part (c) of Lemma 2, we have lp(w,z)I

C

/ i p(z)

+p(z) + lyil + 1w '2 I + lyil

I—n

2

2

+

(4)

for z E D— (because p(z) < 0). The last expression on the right is a locally integrable function in w = (yi, w') E by Lemma 5 (with p = 1/2 and q = n — 1). By the dominated convergence theorem, Kf is continuous up to

MfromD. Derivatives are handled analogously. Suppose f is a function of class

with compact support on M. Let Xi,. .. , Xi" be vector fields on C'2 (not necessarily tangential to M). For z E D,

. . .

involves a sum

of terms, typical of which is

f

(w, z)}kj (w,

where 1 + q < N, where is a differential operator in z or of order q, and where k1 is a smooth function obtained from various derivatives of k. Integrating by parts 1+ 1 times with ô/t9yi yields another sum of terms, typical of which is

f

{f' (w, z)}k2(w,

where is a differential operator in z, and of order at most N + I and where k2 is a smooth function obtained from various derivatives of k1. Since

is of class the estimate in (4) and the dominated convergence theorem imply that . is continuous up to M from D. This completes the proof of Theorem 2. 1 . .

21 Fundamental Solutions for the Tangential Cauchy—Riemann Complex on a Convex

Hypersurface

As mentioned in Chapter 18. there does not exist a solution to the equation (9MXMK = [s], except for very specialized M. However, in this chapter, for a strictly we shall construct a solution to the equation t9MXMK = convex M, modulo a kernel which as an operator acts nontrivially only on forms of bottom and top degree. We shall then abuse the notation and call K a fundamental solution for the tangential Cauchy—Riemann complex. We present two fundamental solutions for the tangential Cauchy—Riemann complex. The second solution will be derived from the first and it will play a key role in the local solution to the tangential Cauchy—Riemann equations in the same way that the Bochner—Martinelli kernel plays a role in the local solution to the Cauchy— Riemann equations. For a form f E the first solution to the tangential Cauchy—Riemann u f will be constructed as the boundary values of terms of the equations form K([M]°" A f) where K is a kernel of the type introduced in Chapter 20.

Since these kernels are ainbiently defined, it will be convenient to use the extrinsic definition of the tangential Cauchy—Riemann complex presented in Section 8.1. The current [M]°" A f is well defined only if f is an ambiently defined form. However, as mentioned in Section 8.1, if F1 and F2 are two ambiently defined forms, with {F1 }tM = f = {F2}tM on Al, then [MJ°" AF1 = [Al]°" AF2. Therefore, for f E we can unambiguously define [M]°" Af to AF where F is any ambiently defined (p, q)-form with FtM = on M. be

f

21.1

The first fundamental solution for the tangential Cauchy—Riemann complex

Let us first summarize the results from Chapters 19 and 20. In Chapter 19, we constructed the Bochner—Martinelli kernel B, which is a fundamental solution

312

The first fundamental solution for the tangential Cauchy—Rlemann complex

313

for the Cauchy—Riemann complex. Suppose M is the smooth boundary of a with the induced boundary orientation. For any element domain D in we showed that {B(IM]°" A f)}tM has continuous nontangential fE These extensions are — {D }). extensions to M from D and D+ (= respectively. We also showed the following jump denoted B(f) and relation holds

B(f) = ftM =f

on

M

In Chapter 20, we defined the H, Ht, L, and Lt kernels for a strictly convex domain D with smooth boundary M. We also showed that there is a neighborhood

U of M such that

0H—L-B onMxIF H

kernels

on

have smooth coefficients on the set M x {U fl D+ }. Therefore, we and may apply the above equations to the current EM]°" A f. Using the equation O{[M)°" A f} = —[M]°" A OMf and the homotopy equation from part (b) of Theorem i in Section 18.2 (with r 0 and s = —2), we obtain

A f)} + H([M]°" AOMI) = L([M]°" Al) — B([MJ°" Al) (la)

on D and Af)}+Ht([M]°" AOMI) = Lt([M]°" Af) —B([M]°" Af) (ib) From Theorems 1 and 2 in Section 20.4, we can take boundary on U fl we set values of the terms appearing in the above equations. For f E

H(f) = extension of {H((M]°" A f)}tMJD- to M

Ht(f) = extension of

L(f)

A f)}tMIUflD+ to M

extension of {L([M]°" A f)}tMID- to M

(Lt)+(f) = extension of

A f)}tMIUflD+ to M.

z) and Ht ((, z) From the proof of Theorem i in Section 20.4, the kernels are locally integrable in ( E M for each fixed z E M. Therefore

H(f)(z)

=

Ht(f)(z)=

{J(EM {J(EM

for z

[H((, z) A

M

} LM

forz€M. tM

Tangential Cauchy—Riemann Complex on a Convex Hypersurface

314

The kernels L((, z) and Lt((, z) are not integrable in (for each fixed z E M. Therefore, these boundary values only exist in the sense described by Theorem 2

in Section 20.4. In particular, the above integral expressions with H replaced

by L do not make sense for z E M. For this reason we have used the — and + superscripts to denote the boundary values of L([M]°" A f)ID- and Lt([M]oI A The operator it'!. Therefore

respectively. = tM o 0 involves only vector fields that are tangential to A

extends

from D to M as OM{H(f)}. Similarly A

to it'! as OM{Ht(f)}. If we apply tpj to equations (la) and extends from (lb), subtract the result, and then use the jump relation = f, we obtain

B(f)

on It'!.

From Lemma 3 in Section 20.3, we have L—(f) = 0 unless f is a (p,0)-form and (Lt)+(f) = 0 unless f is a (p, n — 1)-form. Therefore, except for forms of is a fundamental solution bottom and top degree on M, the operator K = For this reason, we call K a fundamental solution for 8M. From for Theorem i in section 20.4, the operator K is regular (i.e., K sends compactly supported smooth forms to smooth forms). We also have Kt = —K and so K is self-adjoint up to a sign. Therefore, K is a biregular fundamental solution. We summarize the above discussion in the following theorem. THEOREM I The operator Suppose M is a smooth, strictly convex hypersurface in Ht equation is a biregular kernel that satisfies the following operator K = H— on

I=

—

This operator equation has the following special cases: (a)

For

f f=

+

L).

The first fur4amenwJ solution for the tangential Cauchy-Riemann complex

Furthermore, given any positive integer N and any compact positive constant C such that

IK(f)icN(MnK)

for

set

315

K, there is a

ICN(MnK)

allfED'(MflK).

appears in [He3J. This theorem without the The operator equation stated in the above theorem can also be stated in terms of currents as = —COMXMK) + (Lt)+

—

L.

0, (See part (c) of Theorem I in Section 18.2 with m = n — 1, d = 1, r and s = —1). We now state some easy corollaries to the above theorem. The

first corollary (which follows from (a) above) provides a global solution to the equation OMti = f except at the top degree. COROLLARY I

Suppose M is a smooth, strictly convex hypersurface in C'. If f E

for

1

If M is compact, then the above corollary holds with Now let us examine part (b) of Theorem 1. 1ff E

replaced by with '9Mf = 0, then

f = —L(f). For z E D, we have —L([M]°" A f)(z) = ([M]°", L((, z) A

f

[L(çz)

A

CE M

Since u((,z) is indethat L = E(u) where u((,z) z) shows that L((, z) is pendent of z, an examination of the formula for holomorphic in z E D, for ( E M. This yields the following corollary. Recall

COROLLARY 2

Suppose D = D

M in C't. 1ff e

is a strictly convex bounded domain with smooth boundary with = 0, then I has a holomorphic extension

to D given by

F(z) = —L([M]°" A f)(z) = J(EM

z) A

f(()]

The Bochner—Martinelli kernel also provides the holomorphic extension of a CR function as shown in Theorem 1 in Section 19.3. Furthermore, this theorem holds without any convexity assumption on D. The point of this corollary is that

316

Tangensiol Cauchy—Riemana Complex on a Convex Hypersurface

with the additional convexity assumption on D, we have a reproducing kernel for holomorphic functions (namely L((, z)) which is holomorphic in the variable z (this is not true for the Bochner—Martinelli kernel). We also note that on the unit sphere {( = l}. the defining function for M is p(() = K12 — 1 and so u((, z) = (. In this case, L = E(u) is the Szego kernel which represents the L2-projection onto the space of square integrable CR functions on M (see Krantz's book [Kr]). Finally, we take a look at the case of top degree (part (c) in Theorem 1). Suppose M is the boundary of a bounded strictly convex domain D = Every form in (M) is In this case, some other compatibility condition replaces 8Mf = 0 in order to solve the equation OMU = f. For motivation, let us suppose the equation OMu = E eP.n—t(M) has a smooth with solution u. Suppose g = 0. Then

D.

/

(f,g)M = (OMU,9)M =

=0. The second equation follows from the integration by parts formula given in Lemma 6in Section 8.1. Ii follows that a necessary condition for solving the with equation t9Mu = is the condition (/,g)M = 0 for all g (9Mg = 0. Since CR functions on M extend holomorphically to D,we can replace this condition by requiring (f,g)M = 0 for all g with

f

Og = 0. This necessary condition is also a sufficient condition for the existence of a solution to the equation OMU = on M, as the next corollary shows.

f

COROLL4RY 3

Suppose D is a bounded strictly convex domain in C' with smooth boundary The equation M. Suppose f = f has a smooth solution if = Ofor all g £"P°(D) with if and only u =0 on D.

PROOF The necessity has already been discussed prior to the statement of the corollary. For sufficiency, we note from Theorem 1, part (c),

f=

+

(2)

Now Lt = E(u2) and u2((, z) = (Op(z)/c9z) is independent of(. Therefore, the fl coefficients of Lt((, z) are holomorphic in (near M for each fixed z E U}. By Theorem I in Section 19.3 (or Corollary 2 above) Lt((, z) extends to a form G((, z) whose coefficients are holornorphic in ( D. Therefore, for each

A second fundamental solution to the tangential Cauchy—Riemann complex

fixed z E

317

fl U} we have

Lt([M]°" A f)(z) =

-

f

z) A f(()

CE M

=-f

(EM

=

=0 where the last equation follows from the assumed compatibility condition on 1. Therefore = 0, and from (2) we see that u = K(f) is a solution to

the equation liMe =

21.2

f.

U

A second fundamental solution to the tangential Cauchy—Riemann complex

In this section, we construct a simpler fundamental solution for the tangential Cauchy—Riemann complex on a strictly convex hypersurface. This fundamental solution involves only one kernel instead of the two kernels H and Ht that make up the fundamental solution described m the previous section. Let us start by reviewing the notation. We suppose that our convex hypersurface Al divides into two open sets D and D+, where D is convex. Let p be the defining function for D, i.e., D = {(; p(() <0}. As in Chapter 20, we set

u'((,z)

= = lip(z) = ((—z).

We have already defined the kernels H = E(u', u3), L = E(u') and we have shown that their transposes are Ht = E(u2, e3) and Lt = E(u2), respectively. As mentioned in Section 20.4, the H and Ht kernels are locally integrable on M x M and so in this section we shall denote their tangential pieces on M x M by the same symbols H and Ht. That is, on M x M, we set

H = {E(u',u3)}tMXM Ht = {E(u2,u3)}tMXM.

Tangential Cauchy—Rie.nann Complex on a Convex Hypersurface

318

The L and Lt kernels are not locally integrable on Al x M. In Section 20.4, we defined the boundary value operators L and (Lt)+. We now define the kernel that forms our second fundamental solution for the tangential Cauchy—Riemann complex. This kernel is denoted R for Romanov and is given by

R= {E(11',u2)}IMXM. THEOREM I

Suppose Al is a smooth, strictly convex hypersurface in Ctm, n 3.

The R

kernel is a biregular fundamental solution for the tan gential Cauchy—Riemann

complex on M x M. As operators on

oR+R0OM

—

L.

If N is a nonnegative integer and K is a compact subset of M, then there is a positive constant C such that IR(f)IcN(K)
for f E This theorem without the

appears in [He3].

The content of this theorem is that the kernel R can take the place of the kernel K m Theorem 1 in the previous section along with its corollaries. In terms of currents, the statement of the theorem reads OMxMR

—

(Lt)+

+ L = —[s].

(See part (c) of Theorem I in Section 18.2 with m = n S

(1) —

1,

d=

1,

r = 0,

= —1).

PROOF

In view of part (a) of Lemma 1 in Section 20.3, we have

Reversing the roles of ( and z yields

Therefore, the kernels H, Ht, R, L, I) and the kernel E(u',u2,u3) are smoothly defined on the set M x M — From part (b) of Theorem 1 in Section 20.2, we have

OMXM{E(u1,u2,u3)}tMXM =H_Ht_R onMxM—&

(2)

Our goal is to show that the singularity of E(u',u2,u3) is mild enough so that in the sense of currents. For once this is done, this identity holds across

A second fundamental solution to the tangential Cauchy—Rlemann complex

319

equation (1) follows by applying 0M,< M to both sides of equation (2) and then using the equation 49MxM{H —

Ht} =

—[A] + (Lt)+ —

which is the current equation associated to Theorem 1 in the previous section.

To show that (2) holds across A, we need the following lemma. LEMMA 1

The kernels R, H, Ht, and E(u', u2, u3) are currents with locally integrable coefficients on M x M. (b) Given a compact set K C M, there is a positive constant C such that for each 0 and z E K (a)

Ce2. J IE(u',u2,u3)((,z)I (EM

Let us assume the lemma for the moment and show that equation (2) holds

across A. This will complete the first part of Theorem 1. be a smooth function on M x M such that For e > 0, let (/. The

function

can be

\f 1

chosen so that if D

is

any first-order derivative, then

where C is a positive constant that is independent of e.

Since

vanishes near A, we have from (2)

A E(u',u2,u3)

= —

on

Ht

—

R)

(3)

M x M. From part (a) of Lemma 1, we have

x€HtI_*Ht R,

h—'

as e '—. 0, where the convergence holds in the weak topology on the space of

currents on M x M. Since differentiation is a continuous operator in the space of currents with the weak topology, we have OMXM{X(E(&,u2,U3)} i_+OMXM{E(U',u2,U3)}

Tangential Cauchy—Riemann Complex on a Convex Hypersurface

320

as

'—p 0.

Moreover, the estimate

together with the estimate given in part (b) of Lemma 1 implies

A E(u',u2,u3) '—p0 0, in the sense of currents. By taking limits of equation (3) as e 0, we see that equation (2) holds on all of M x M, as desired. This completes on R(f) which we will the proof of Theorem 1 except for the

as

establish after giving the proof of Lemma 1.

I

The proof of this lemma is much like the proof of Theorem i in Section 20.4. In fact, we have already shown in the proof of PROOF OF LEMMA 1

that theorem that the kernels H and Ht have locally integrable coefficients on

ivi x M. So we turn our attention to the kernels R and E(u', u2, u3). An examination of these two kernels leads us to estimate a term of the form

K"

k((,z)

-

—

where

q,rl, q+r+8=n. For the R kernel, we have s =

0

and q + r =

n.

The function k involves

coefficients of the differential form

(Op(() .d((_z))

.d((_z))

A

which vanishes when (= z. Therefore, we have

k((,z) = O(K—zl). For

the kernel E(u',u2 ,u3), we have s 1. In this case, k involves coeffi-

cients of the differential form

(Op(()

.d((_z))

A

.d((_z)) A(((_z).d((_z))

which vanishes to second order on {( = z} . Hence, we have

k((, z) = O(I( —

z12).

A second fundamental solution to the tangential Cauchy—Rlemann complex

321

Since the R and E(u', u2, u3) kernels have coefficients that are smooth on MxM— we only need to estimate IK((, z)i for (, z in a neighborhood in M of a fixed point z0 M. We use the change of variables w = 'I',, (() given in Lemma I in Section 20.4. This lemma implies that for a given ZO M, there is a neighborhood U of z0 in such that for each z U fl M, the map :U is a diffeomorphism onto its image; and that is an open neighborhood of the origin in the

copy of given by {w = (w1,. .. we see that we must estimate 1

=

ki(w,z)

\ /

Rew1 = 0}. Letting

,

—t

(p(w,

(w)

—

zi 2

where

ki(w,z) = k(W'(w),z) Op(W;'(w))

/ — Z

a

—

p*(w,z)= As in Section 20.4, we give the coordinates w = (yi,w') where yi E IR and w' Here, we use the following notation. Suppose g x

C is smooth. We say that

g(w, z) = 0(w)' provided g is a polynomial of degree j in the real components of w = (Yi , w') E ' over the ring of smooth functions defined on If g is real valued and

ifg=0(w)', then we say that g

wi'

0 such that g(w, z) €IwIi. One of the differences between the proof of Lemma 1 and the proof of Theorems 1 and 2 in Section 20.4 is that here, the point z is always in M, whereas in Section 20.4, the point z lies to one side of M. In our present setting, this has the simplification that p(z) = 0 and so 0 and hence provided there is a uniform

= z. For z EM and w W;'(w)

we have — zi

Iwl.

This is a key observation which will be used in the proof of the following

analogue of Lemma 2 in Section 20.4.

Tangential Cauchy—Riemann Complex on a Convex Efypersurface

322

LEMMA 2

There is a neighborhood U1 of zO in M and there is a neighborhood U2 of the origin in such that the following hold: (a)

Ifk((,z) = 0((— z)3 for(,z EU1 then k1(w,z) =

= 0(w)' for z E U1 and w e ('2. (b) IW;'(w) —

z E U and wE U2.

z EU1 andw EU2. (d) p(w,z) = 0(w)2 for z EU1 andw EU2. (e) There is a Constant > 0 such that (c)

and

e(iwhl2

p(w,

+ yiJ)

z)i e(iw'12 + )YI

for z EU1 and w = (y1,w') EU2. PROOF The proofs of parts (a)—(c) of this lemma are similar to the proofs of the corresponding parts of Lemma 2 in Section 20.4. As already mentioned, W is the copy of is a local diffeomorphism on an open set U C and given by {Rew1 = z for z M, there is an open OL Since neighborhood U1 of zO in M and an open neighborhood U2 of the origin in with 'P;'{U2} c Un M for each z E U1. Parts (a) and (b) now follow about the origin. from a Taylor expansion in w E For part (c), we first note from part (b) of Lemma 1 in Section 20.3 that if ( and z belong to M = {p = 0} then I

-

- z)}

Re {

With the roles of ( and z reversed, we obtain

ôp(z)

z)}

—Re {

-

for and z E M. Therefore, part (c) follows by letting (= using part (b).

Part (d) also follows from part (b) and the equation

((—z)

öp(z)

—

(Op(()

=0((_z)2.

ôp(z)

and then

A second fundamental solution to the tangential Cauchy—Riemann complex

323

The first estimate in part (e) follows from the estimate Rep(w, z) fIwI2 in part (c) together with the fact that Imp(w, z) = y'. For the second estimate in part (e), we first note from part (c) that

w

U2. From part (d), we have

IImp(w,z)I — IIm{p(w,z) _p*(w,z)}I —

C(Iw'12 + IyiI2) —

Clw'12

provided vt and 1w'! are suitably small, where C is some uniform positive I

constant. Therefore, we obtain Ip*(w,z)I max

—

CIwhl2}.

(4)

Now we use an easily established inequality (also used in a similar context in

[GL]): if a,/3,y > 0 then

max{a,j3—'y} (2+

(a+$).

(5)

a + 8 and this inequality follows by dividing If a then 2a + — through by 2+ ('y/a). On the other hand if a B — -y, then <

-

= as

desired.

If we use inequality (5) with a =

€Iw'12, i3

and

y

CIw'12,

then (4) becomes

Ip*(w,z)I

(2+ £) ' (€w12+

the second inequality in (e) is established. This completes the proof of I Lemma 2. and

Now we return to the proof of Lemma 1. To show the R kernel has locally integrable coefficients, we must show that K1 (w, z) is locally integrable in

Tangential Cauchy—Riernann Complex on a Convex Hypersusface

324

w = (y1.w') E

in this case

Ki(w,z)=

ki(w, z)

q+r=n.

z)t'

p(w,

where k1 (w, z) = (9(w) in view of part (a) of Lemma

2.

(6)

Using part (e) of

Lemma 2. we obtain

IK(w,z)I

CIwj (Iw'V +

for z E U1 w EU2

C - (IwII2 +

C is a uniform positive constant. The right side is a locally integrable in view of Lemma 5 in Section 20.4 (with function of w = (yi,w') E q = n — (1/2) and p = 1/2). This completes the proof that the R kernel has where

locally integrable coefficients.

For E(u', u2, u3), we must examine the term

Ki(w,z)=

kj(w,z) p(w,

*

(iv, Z)rIWZ—1 (w) —

q+r+Sfl.

28

z) = O(1w12) in view of the discussion at the beginning of the proof of Lemma 1. Lemma 2 yields the estimate This time, k1

(w,

— (Iw'12

+ C

+ jyi

s 1 and q + r + s = n, the right side is locally integrable in w by Lemma 5 in Section 20.4. (yi,w')

Since

Part (b) of Lemma 1 also follows from the above estimate and by integrating and then with respect to w', i.e., the right side first with respect to

f [ J

J

dyidv(w') (jw'j2 +

[ —

dv(w')

J c1€2

where C1

is

a uniform positive constant. The last inequality follows from a

standard polar coordinate integral calculation in the proof of Lemma 1. I

This completes

A second fundamental solution to the tangential Cauchy—Rlemann complex

325

As mentioned earlier, Lemma 1 shows that equation (2) holds across from which the first part of Theorem I follows. The only thing remaining in the proof of Theorem 1 is to establish the via estimate on R(f). Pulling back the integral appearing in R(f)(z) to we see that we must examine a term of the form

J

Kl(w,z)fi(w,z)dv(w)1

(7)

)

where K1 is given in (6) and 11 (w, z) is a coefficient of the pull back of the are tangential vector fields form f via the map ( = W; '(w). Here, Xi,... , can be established by differentiating under the to M. The desired integral sign which is valid provided we show

is dominated by a locally integrable function in w = (yr, w') E uniformly in z E U1 C M. Since we already know that K1 (w, z) is locally integrable in (Yi, w'), it suffices to show that differentiating K1 (w, z) with a tangential vector field in z does not worsen the estimates. As with the proofs of Theorems 1 and 2

in Section 20.4, the key idea is to note that if g is a smooth, complex-valued function with g(w,z) = 0(w)3 for w E and z E M for some j 0, z)} = 0(w)3 provided then also is a tangential vector field to M. From parts (a) (with j = 1), (c) and (d) of Lemma 2, we obtain

= 0(w) = 0(w)2

=

z)} = Xz{p*(w, z) — p(w, z)} + X2{p(w, z)}

= 0(w)2.

(8)

Note that Lemma 2 only holds for z E M and therefore the above estimates only hold for vector fields that are tangential to M. Now we show that the estimates are no worse when we differentiate K1 (w, z) with a tangential vector field. For example, we have

z))}I = < —

z)}IIp(w, C1w12

(IwhI2

+

(IwhI2+

(from Lemma 2 and (8))

326

Tangential Cauchy—Riemann Complex on a Convex Hypersurf ace

where C is some uniform positive constant that is independent of z E U1 C M and w (yi, w') E U2 C Wn—I This is the same estimate that is satisfied by Repeating the above arguments we can establish the following: if Ip(w, Xi,. , are tangential vector fields to M. then there is a uniform positive constant C such that

<

z EU1 CM and w

(y1,w') e

C (Iw'V +

Since the right side is locally inte(by Lemma 5 in Section 20.4), we can differentiate under grable in w E the integral sign in (7) and the proof of the desired C1V -estimate is complete. for

22 A Local Solution to the Tangential Cauchy—Riemann Equations

In Chapter 21, we constructed the R kernel which is a biregular fundamental solution for the tangential Cauchy—Riemann complex on a strictly convex hypersurface. This kernel is analogous to the Bochner—Martinelli kernel which is a biregular fundamental solution for 0 on C's. In Theorem 1 in Section 20.3, we used the Bochner—Martinelli kernel together with a kernel of Henkin to construct a solution to the 8-equation on a strictly convex domain in In this chapter, we shall use a similar procedure with the R kernel to construct a local on a strictly solution to the tangential Cauchy—Riemann equations (9)14 u = pseudoconvex hypersurface. As with global solvability, there is an obstruction to the local solvability of the tangential Cauchy—Riemann equations at top degree. In Chapter 23, we will discuss necessary and sufficient conditions for local solvability at the top degree. In this chapter, our goal is to prove the following theorem of Henkin's.

f

THEOREM 1

(See [He3J.) Suppose M is a smooth real hypersurface in n 3, and let z0 be a point in M. Suppose that M is strictly pseudoconvex at ZO. Then there is a local neighborhood basis C of open sets in M about z0 with the following property. Suppose w E C and let f be a (p,q)-form, 1 q n —2 which is C1 on with 0Mf =0 on w, then there exists a (p,q — 1)-form u which is of class C' on w with OM'U = on w.

f

The proof also exhibits a solution u by integral kernels. By a local neighborhood basis about ZIJ, we mean a collection of open sets C in M about with the property w

{zo}.

Note that we are not shrinking the set on which we solve the tangential Cauchy—

Riemann equations. That is, if f is OM-closed on

C, then there is a solution

327

A Local Solution to the Tangential Cauchy—Riemann Equations

328

u for the equation OMU = I on all of w. However, we are making no claims as to the regularity of the solution u at the boundary of w. More will be said about boundary regularity in Chapter 24. We should point out that not just any local neighborhood basis in M about zO will satisfy the above theorem. Such a neighborhood basis must be specially

constructed as we shall do below. This is analogous to the situation with 0 Using Theorem 1 in Section 20.3, we can find a local on domains in neighborhood basis consisting of balls in on which we can solve the 0equation. The reader familiar with the theory of several complex variables knows that the 0-equation cannot be solved on an arbitrary open neighborhood Such a set must be a domain of holomorphy. On of a given point zO in a strictly convex hypersurface, it is unknown how much flexibility one has in constructing the neighborhood basis on which the tangential Cauchy—Riemann equations can be solved. The rest of this chapter is devoted to the proof of Theorem I. Since our analas in Theorem 1 ysis is local, we may choose holomorphic coordinates for in Section 10.3 so that M is a strictly convex hypersurface (near the origin) in so that the given More precisely, we choose coordinates (z1, z') for point zO is the origin and so that a defining function for it'! is

p(z) = Imz1

where h : P x

—

h(Rezj,z')

P is smooth with h(O) = 0, Dh(O) = 0 and we assume

the real hessian of h at the origin is negative definite. We set D = {z p(z) > 0}. Note that if U is a small p(z) <0} and {z E enough ball centered at the origin then U fl D

is convex.

As in Chapters 20 and 21, we let u'((,z) =

and u2((,z) =

(Op(z)/Oz) and we form the kernels

L= Lt

{E(u2)}

R = {E(u',u2)}tMXM. We restate the fundamental identity for these kernels (1)

(as operators on fl U)). Now we define our local neighborhood basis for Theorem 1. For A > 0, let WA

Since h(0) =

0,

Dh(0) =

= {z

M; Imz1 > —A}

= {z

M; h(z) > —A}.

0

and since the real hessian of h at the origin

is negative definite, the diameter of

is proportional to

provided A is

A Local Solution to the Tangential Cauchy—Riemann Equations

329

suitably small. So the collection

C={WA; A>0} a local neighborhood basis for the origin in M. — We fix a small A > 0 and show that the equation OMU = can be solved on where f is a smooth (p,q)-form on I q n — 2, with OMf = 0 is

/

on

Define

u3((,z)=(l,0...0)E C't. We

form the kernels

E123

= {E(u',u2,u3)}tMXM

E23 = {E(u2,u3)}gMXM E13

= {E(v',u3)}tMXM.

z) = 0, we have E(u3)

Since Note that

and

E13

= 0.

is self-adjoint up to a sign and Eb =

E13.

The

kernels E123,

are smoothly defined on the set

V=

E {MnU} x {MnU};

This is because u3((,z) ((—z) =

— z1

z1}.

and because u'((,z) ((— z) and

u2((, z) ((— z) are nonvanishing on M x M — Therefore, these kernels are biregular on the set V. This means that if U1 and U2 are open sets in M with U1 x U2 C V. then these kernels represent continuous operators from to Moreover, these kernels extend to operators from (compactly supported currents on U1) to ((12). From part (b) of Theorem I in Section 20.2, we have

onV.

(2)

In Lemma 3 in Section 20.3, we saw that the L kernel acts nontrivially only on forms of bidegree (p, 0) and Lt acts nontrivially only on forms of bidegree (p, n—i). The following lemma describes the analogous behavior for the kernels E13 and LEMMA 1

Suppose Ui and U2 are open sets in M with

U1

x U2 C V. Suppose g is a

compactly supported current of bidegree (p, q) on U1. (a)

If q

I, then E13(g) = 0

(b)

If q

n — 1, then F223(g) = 0 on U2.

on U2.

A Local Solution to the Tangential Cauchy—Riemann Equations

330

As with the proof of Lemma 3 in Section 20.3, note that u1 and u3 are holomorphic in z and so the degree of z) in d( is n — 2 (see the formula for E(u', u3) given in Section 20.1). For 2 e U2, we have PROOF

E13(g)(z)

=

f

CE Ut

=0 unless g is a form of bidegree (p, 1). The above integral formula is well defined provided g is a form with continuous coefficients on U1. If g is a more general current, then the above integral gets replaced by the pairing ) keeping in mind the fact that E13 has smooth coefficients on U1 x and the same conclusion holds.

Part (b) follows by the same reasoning and by noting that the degree of E23(C,z) in d( is zero. I To solve the Fix a smooth (p, q)-form f defined on with OMf = 0 on equation we start by applying the R kernel to the current = on where is the characteristic function on the set Since R is a bigregular

f

kernel on {MflU} x {MflU}, R(xAf) is a well-defined current on MflU. in fact, since R has locally integrable coefficients (see Lemma 1 in Section 21.2) and since xAf is bounded, R(XAI) is a form with continuous coefficients on = 0 and =0 UnM. Also note that if I q n—2, then in view of Lemma 3 in Section 20.3. From (1), we obtain +

xAf =

= 19M{R(XAf)} — The second equality uses the fact that This follows from the equation OMX.\ =

A f).

=

=

as a current on M. (by Stokes'

theorem). — It is instructive to compare the above equation with the analogous 0-equation kernel appearing in the proof of Theorem 1 in involving the Section 20.3 In the proof of that theorem, we used the equation OH = L — B to rewrite the term B([Mj°" A f) as —0{H(LM}°" A f)}. Here, we want to use With the equation (2) to show that R([OwA]°" Al) belongs to the range of

help of part (C) of Theorem 1 in Section 18.2 and using —IOWA]°" A OMf = 0, we obtain

Al) = aM on

zE

Al)

If K is either E123, E23, or E13, then K(fi9wA]°" A f) is smoothly = and ( E OWA then Imz1 > —.\ and For if z E 0. Hence, the denominators of K(ç, z) are nonvanishing for so —

defined on and

Af)}+E13([OWA]°"

A f} =

and ( E OWA.

A Local Solution to the Tangential Cauchy—Riemann Equations

If f is a (p,q)-form with I

q
3,

331

A f is a current of

then

A f) =

0

A f) =

0

in view of Lemma 1. Combining this with the previous two equations, we with obtain a solution to the equation 8M u = f on

u(z) = R(XAf)(z) =

f

A

—

A f(()

[E123(cz) A

-J

f

(3)

B(xDf) + H([aDl°' A 1) of

(Compare this equation with the solution

the equation Ou =

f)(z)

on a convex domain D in

given in Theorem 1 in

Section 20.3.)

Theorem I also states that the equation u = f can be solved when f is a form of bidegree (p,n — 2). In this case, [OwA]°' A f is a current of and therefore Lemma I cannot be used to show that the term E23([awx]°" A f) vanishes. However, the next lemma uses an approximation 0 and therefore the above form argument to show that E23([ÔWA]°' A f)

bidegree (p, n —

1)

u solves the equation (p,n —2).

=

f

even when I is a form of bidegree

on

LEMMA 2

Suppose f is a smooth form of bidegree (p, n — 2) on

with OMf =

0 on

Then A

f) = 0.

First note that the coefficients of E23 ((, z) are holomorphic in ( provided the denominators of E23 are nonvanishing. We are tempted to write PROOF

A f = —ÔM{XAI}

(since ÔMf = 0)

and then integrate by parts with aM to show

=

-

f

A

E WA. However, this argument is not valid because E23(ç, z) is not smoothly defined for (,z E (when (' = z1) and so the integration by parts

with ÔM is not allowed. Instead, we first approximate the coefficients of z) by entire functions of and then we apply the above integration by parts argument. To carry this

A Local Solution to the To,ngentlal Cauchy-Rknsann Equations

332

out, we fix z

Since

f has bidegree (p, n — 2), we have A f)(z)

=

J

A

where is a form of biciegree (n — p,O) in (. The coefficients of are holomorphic in ( provided the denominators

OP(Z)(() are nonvanishing. For fixed z

E

and

((1—zi)

let

A={(€UCC"; Im(1 =

(so Lmz1 > —A) then (i — 0. Moreover, part (a) of Lemma 1 in Section 20.3 with the roles of z and ( reversed

implies that if (Op(z)/Oz) ((— z) = 0 then either ( = z or p(() > p(z). z) then A and z WA C M (i.e., p(() ( 0 = p(z) and ( then the coefficients of z) 0. Therefore if z E (Op(z)/Oz) ((— are holomorphic in (for (in a neighborhood of A in

So if (

Since A is a convex set, A is polynomially convex. Hence, a function which is holomorphic in a neighborhood of A can be uniformly approximated by a sequence of entire functions (see (Hol). It follows that for fixed z WA, can be uniformly approximated for ( A by a sequence of (n — p, 0) forms G3((), j = 1,2,..., whose coefficients are entire functions of ( C'2. Since Ow,, C A, we have

E2(() A f(()

= lim J = urn J

G,(() A f(() A

f Af has bidegree (n, n —2). Since the fact that G3 is entire and OM f = 0, the last limit vanishes and the proof of Lemma 2 is I complete. As

mentioned earlier, this lemma implies that the form u given in (3) is a

when f is a OM-closed (p, n — 2)-form. solution to the equation OMU = f on The proof of Theorem 1 is now complete. I Let us examine the above analysis in the case when f is a smooth CR function defined on WA. We shall see that the above formulas yield an integral

kernel representation of the local holomorphic extension of I to the convex

A Local Solution to the Tangential Cauchy—Riemann Equations

333

side of WA C M given in Hans Lewy's CR extension theorem (Theorem I in

Section 14.1).

First, note that if f is a CR fimction on

then (I) yields

= —R([0w.>j°' A 1) — L(XAJ). Using (2), we obtain R([WA]0" A f) = E13([OWA]°" A 1).

So f is the boundary values on WA from D of the function —

Both E13((, z) and

E13([OWA]°' A

f) —

(4)

on WA.

z) are holomorphic in z provided

z)

—

z)

0

and (' z1. Therefore, by part (a) of Lemma 1 in Section 20.3, L(xAf)(z) is holomorphic in z for z E D. By the same reasoning, E13([OwA]°" A f)(z) is holomorphic in z for z E D with Imz1 > —A. Thus, the function in (4) is the holomorphic extension of f to the set {z E z E D and Imz1 > —A}.

23 Local Nonsolvability of the Tangential Cauchy—Riemann Complex

As was seen in Chapter 22, there is an obstruction to the local solvability of the tangential Cauchy—Riemann equations at the top degree. In this chapter, we discuss this obstruction in more detail. The system of tangential Cauchy—Riemann equations at the top degree is no longer an overdetermined system of partial

differential equations. For example, if f is a form of bidegree (n, n — 1) on a real hypersurface in then the equation 0Af U = consists of one partial differential equation and one unknown coefficient function. If M and f are real

f

analytic, then by the Cauchy—Kowalevsky theorem, there is a local solution to the tangential Cauchy—Riemann equations. For some time, it was thought that could replace "real analytic" in the statement of the Cauchy—Kowalevsky theorem. However, in 1957, Hans Lewy [L2] found a counierexample which

we present in Section 23.1. We then show that Hans Lewy's example can be recast in the language of the tangential Cauchy—Riemann complex of the Heisenberg group. In particular, Hans Lewy's example provides an example of the local nonsolvability of the tangential Cauchy—Riemann equations at the top degree. In Section 23.2, we consider a more general real analytic, strictly pseudoconvex hypersurface in and we present Henkin's criterion on a smooth form I for the local solvability of the tangential Cauchy—Riemann equations

ÔMu=f.

23.1

Hans Lewy's nonsolvability example

Give R3 the coordinates (xI,x2,y2). Let z2 = x2 + iy2 E C. Define the following differential operator on 1R3:

—8- .8 L=

334

Hans Levy's nonsolvabiity example

335

THEOREM 1

(See [U].) Suppose f is a continuous real-valued function depending only on in some x1. If there is a C' function u of (x,, x2, 112) that satisfies Lu = neighborhood of the origin, then f is real analytic at x1 = 0.

f

that is not real analytic, then Hans Lewy's If f is a smooth function of theorem implies that the equation Lu = has no locally defined C' solution.

f

Therefore, the Cauchy—Kowalevsky theorem does not hold with "real analytic" replaced by "C°°".

We follow the presentation given in [Fo). Suppose the equation Lu = has a C' solution u defined on the set {(x1, z2); lxii
f

V(xi,r) by

V(x,,r)

=

f

u(x,,z2)dz2.

By Stokes' theorem, we have

V(x,,r)= f Z2I

I = J

I
I

=

2iff

Therefore r2lr

ÔV(x,,r) ar

=21] f

=2

J

an

dz2

—(x,,z2)r—.

1z21=r

we let s =

and we think of V as a function of x1 and s, which is C' on the region {ixi I
r2

—

—

8V dr or ds

9V1 Or 2r

=

[

i

j

Ou

uz2

dz2

Local Nonsolvabilliy of the Tangential Cauchy-Rlemwui Complex

336

We use

f to

the equation Lu

obtain

[Ott J

9V

tdz2

J—

1z21r

I

.Ov (1)

We set F(si)

=

f(t)dt and we U(xi,s)

Equation

(1)

implies that U

define

= V(xi,s) +2irF(x1).

is holomorphic

as a function of the complex

variable

w = x1 + is in the region {lxil < R and 0 < $ < R2}. From the definition of V. we have V(xi 0) = 0. Therefore, U is continuous up to s = 0 (from $ > 0) and U(xi,0) = 2irF(x1) E R. By the Schwarz reflection principle, U holomorphically continues to the region {lxi I
is also a real analytic function of x1 for lxii < R, as desired.

I

f

The vector field L in Hans Lewy's counterexample also arises from the tangential Cauchy—Riemann complex of the Heisenberg group M = { (z1, z2) E

Theorem 3 in Section 7.2, a basis for H°"(M) is — ir : C2 IR x C by given by L = Define the L. Therefore, the ir(z1,z2) = (xl,z2) where x1 = Rez1. We have equation Lu = f on R x C is equivalent to the equation Lu = f on M where ü(xi + 21z212, Z2) = u(xi ,z2) and f(xi + iiz2i2, z2) = f(xi, z2). We claim the equation Lu on M is equivalent to OM{u} = To see this, we first note that p(zi,z2) = 1z212 — Imz1 is a defining function for M. In order to compute the tangential projection map tM and the 0M C2;hnz1

From

/

operator, we need the following dual vector field N to Op:

From Lemma 2 in Section 8.1, we have OMU = N.i(OpA&ü)

= (I Z2 '.

Oil

i Ou'\ + —— I

2tTh2)

— I \.

41z212+l

and

= —

A

i

-

—

4iz2i2+1

Henkin 's criterion for Local solvablWy at the top degree

Therefore, the equation 8MÜ =

is

337

equivalent to the equation -

=

—

I

as claimed. So Theorem I implies the following local solvability (or nonsolvability) result for the tangential Cauchy—Riemann complex on the Heisenberg group. COROLLARY 1

f

Let M = 22) E C2; Imz1 = 1z212} and let be a smooth, real-valued function of Rez1. Suppose there exists a smooth function u defined on a neighborhood of the origin in Al such that u = f must be real analytic in a neighborhood of the origin. The Heisenberg group {hnzi = 22 I2} is biholomorphic to the unit sphere via the Cayley transform: 4': C2 C2, 4'(w1, W2) = 22) with 21 =

.11—wi 1 I

W2 22

I + Wi

In these coordinates, 4' takes the unit sphere in C2 biholomorphically to the Heisenberg group. Therefore, the above nonsolvability example on the Heisenberg group can be carried over to a nonsolvability example on the unit sphere. Another example of the local nonsolvability of the tangential Cauchy—Riemann equations on the unit sphere will be given at the end of the next section.

23.2

Henkin's criterion for local solvability at the top degree

In this section, we discuss local solvability of the tangential Cauchy—Riemann complex at the top degree for a general real analytic strictly pseudoconvex hypersurface M in n 2. As in Chapter 22, we may use a local biholomorphic

change of coordinates and assume M is a strictly convex hypersurface in an open neighborhood U of a given point Zo in We assume M is defined by {z E U; p(z) = 0} and that M divides U into two open sets = {p > O} and <0} with U convex. As in Section 20, we construct the kernels L, {p = Lt, H, and Ht. Suppose f E and suppose E D(MflU) with = I in a neighborhood of Zo. From part (c) of Theorem I in Section 21.1, we have

01 =

—

Ht)(Of)} +

(1)

338

Local Nonsolvabiluly of the Tangential Cauchy-Rlemann Complex

f

The term (Lt)+(Øf) is the obstruction to solving the the equation = in a neighborhood of z0 in M. This term is analyzed in the following theorem which is due to Henkin [He3]. THEOREM I

Let M be a strictly convex real analytic hypersurface in an open set U in n 2 and let 20 be a given point in M fl U. Let V(M fl U) be identically one on a neighborhood of 20 mM. 1ff is a smooth form of bidegree (n, n — 1) on M fl U, then the equation OMU = has a locally defined smooth solution near 20 Ofl M jf and only is real analytic in a neighborhood of 20 in M.

f

Before we prove this theorem, we show the statement

is real

analytic near 20" is independent of the cut-off function For suppose and are two such cut-off functions. Then v = vanishes in a neighborhood of 20 and has compact support in Mn U. Since the defining equation for M (p)

is real analytic, Lt((, z) is real analytic in z provided (Op(z)/Oz) ((— z)

0.

So (Lt)+(v) is real analytic at a point z Mn U provided

{(EM; Oci(z)

(2)

Since v vanishes in a neighborhood of zo and since M is strictly convex in U, part (a) of Lemma i in Section 20.3 (with the roles of z and ( reversed) implies that (2) holds for z in a neighborhood of 20 in M. So (Lt)+(vf) is real analytic in a neighborhood of 20 in M. It follows that

+

=

is real analytic in a neighborhood of 20 in M if and only if analytic, as claimed. PROOF

is

'(Mn U) over the ring of smooth func-

Since the dimension of

f

tions on M is one, the equation n U) consists = for f E of one partial differential equation. If (Lt)+ is real analytic in a neighborhood of 20, then by the Cauchy—Kowalevsky theorem, there is a solution to the equation OMv = near 20 in M. From (I), we obtain

f=çbf =

(nearzo) —

Ht)(Øf) + v}.

Conversely, let us suppose on some open neighborhood U1 of 20 in M, there is a solution to the equation = f. Choose a smooth cut-off function with compact support in U1 with 0 = I near zo. We have

=

339

Henkin's criterion for local solvability at the top degree

is the boundary values from U± of

near zo on M. Recall that the form

- {(EMnU} J

Lt((, z) A

Also recall that Lt((, z) is holomorphic in (

E

U

for z E Ut We can

integrate by parts with aM to obtain

= Since p is

real analytic, (Lt)+(OMØA u)(z) is real analytic at z E M provided

E M;

ôp(z)

{

((— z) = o} n

=

= 0 near z0 in M, it follows from part (a) of Lemma I in Sec= tion 20.3 that (3) holds for z in a neighborhood of 20 in M. Thus, Since ÔMó

A u) is real analytic near 20 on M, as desired.

I

As an application, we construct examples of local nonsolvability of the tanIn this case, our gential Cauchy—Riemann equations on the unit sphere in hypersurface M is compact and so we set = I. A defining function for M is p(z) = So (ap(z)/az) = The set U is the open unit ball < and the set is the outside of the unit ball {IzI > l}. Suppose I E

For z( > 1, we have Lt([M]°1 A f)(z)

= notation indicates the piece of of bidegree Since f has bidegree (n ,n — 1), the term Lt ((, z) will only contribute dz'.s and di's to the above integral. For zI > 1, we have Recall that the (ii, n — 1) in

dz)

A f)(z) =

For f e

define

(L1 the function If on U by

11(w) =

340

Local Nonsolvabiluty of the Tangential Cauchy—Riemanu Complex

Note that If is holomorphic on the set U = {IwI <

I }.

From the above

expression for Lt([M1ol A f), we obtain

Lt([M1°' A f)(z) =

dz)

((If)

A f) extends from = {IzI > l} to a smooth form on = I } (see Theorem 2 in Section 20.4), the above equation shows that If extends from {lwj < I } to a smooth function on M. Since If is holomorphic on {IwI < l}, the extension of If to M is a CR function. Also

M = {

J

z

note that (Lt)+(f) has real analytic coefficients near E it'! if and only if the If(VJ) is real analytic in w near ZO. This in turn is equivalent to function w If(w) for w near From Theorem I the real analyticity of the function ic in PvI extends in Section 9.1, a real analytic CR function on M near a point to a holomorphic function on a neighborhood of in C's. Therefore from Theorem 1, we obtain the following corollary. COROLLARY I The equation OM u = f can be solved on M

I } near Zo E M if and {z only if the function If extends to a holomorphic function in a neighborhood of

To construct examples of nonsolvability of the tangential Cauchy—Riemann

equations on the sphere near a given point z0, we first find a holomorphic < l} which extends smoothly to M = {IzI = l} but does function f on not holomorphically continue past we can let

For example, if ZO

= (1,0,... ,O) then

where we use the principal branch of the square root defined in the right half plane in C. Next we define for KI =

do A

f(() Thedegreeoffis(n,n—l). is the inclusion map. It follows that

do =

d). Therefore

.

on M. We obtain

If(w) =

(—21r2)

-n JgI=i

(1

—

Henkin's criterion for local solvability at the top degree

After the change of variables (i-i'

341

(n the above integral, we obtain

11(w) 1 —

= The

f

f(() A

d((

- w) A(d(( w).d(( ]

factor of (—1 )" disappears due to the change in orientation of the map

(p-. (. From the formula for L (with z replaced by w). we obtain 11(w) = L([M1°" A f)(w). Since I is holomorphic on the set {IwI < 1}, f = —L([M]°" Af) on {IwJ < 1) by Corollary 2 in Section 21.1. We obtain

If =—! on {IwI < 1). Since / does not holomorphically continue past the point = = (1,0,. . . , 0), neither does If, and so the equation OMU = cannot be

solved in any neighborhood of zo in M according to Corollary 1.

f

24 Further Results

24.1

More on the Bochner—Martinelli kernel

The boundary value result in Theorem 3 in Section 19.2 for the Bochner— Martinelli kernel can be strengthened. If we only assume that f is of class for 0 < < on M, then B([M]°" A f) is of class Ca on D and This can be established by adapting the proof of the corresponding result for 1

the Cauchy kernel in Lemma 2 in Section 15.4. Cauchy's integral formula used in the proof of that lemma must be replaced by the equation

f

= -B([M]°"f)

(EM

=xv-f

(1)

which holds for functions f which are holomorphic on D and continuous up to M (see Theorem 3 in Section 18.2). Here, XD- is the characteristic function of D whose manifold boundary is M. The minus sign in the first equality in (1) results from commuting the current [M)°" with the (2n — 1)-form B((, z). Adapting the computations in the proof of Lemma 2 in Section 9.4 for the Bochner—Martinelli kernel is somewhat tedious since the Bochner—Martinelli kernel is more complicated to unravel than the Cauchy kernel. However, the basic ideas are the same. This result for the Bochner—Martinelli kernel is due to Cirka [C], who a.lso generalized Theorem i in Section 19.3 to the case where the ambient space is a Stein manifold (instead of merely It is interesting to compare the boundary values of the Bochner—Martinelli kernel with the principal value limits of the Bochner—Martinelli kernel. The

latter is defined for f

Bbf(z) = lim EI—.O

342

by

f

{B((, z) A

z E M.

kernel

More on the

343

The following equations hold for f E

B(f) = Bb(f)

—

(2)

= Bo(f) +

(3)

Here, M is oriented by the equation dxD- = —[M}. These relationships are well known from residue theory for the Cauchy kernel. Note that subtracting these two equations gives the key boundary value jump formula B (f) = which is crucial in Chapter 21 for the construction of the fundamental solution for 8M• Equations (2) and (3) are due to Harvey and Lawson

f

[HLJ for the case where f is a function and to Harvey and Polking [HPI for the case of higher degree forms.

We shall sketch the ideas involved in the proof of (2). The proof of (3) is similar. The proof of (2) can be reduced to the case where f is a smooth function by using Lemma 2 as in the proof of Theorem 3 in Section 19.2. We assume that M is the boundary of a bounded domain D. The case where D is unbounded can be handled in a similar way by using cutoff functions as in the proof of Theorem 3 in Section 19.2. Fix Zo E M. If f is a smooth function is an integrable function of ( E M. that vanishes at zo, then IB((, exists by the dominated convergence and Bbf(zo) = So Bbf(zo) theorem. This proves (2) at zo for the case of a smooth function f that vanishes at z0. For more general f, we may write f(() = (f(() — f(zo)) + f(zo). Since we know (2) holds at zo for the function ('—' f(() — f(z0), it suffices to show (2) for the constant function I (() = 1. For z E D and 0, we have

B([M]°'t l)(z) =

-

f

z)

(EM

-

f

B((,z)

(4)

—

where we have set

M; K

< €}. Also set

—

= {(

E

Note that = [St] — [M€], where M( has the same orientation as M has the induced boundary orientation from From (1) with

and

= 0 for z E D. For

instead of D, we have B([8D7]°")(z) = z E D, we obtain

J B((, z) (EM.

=

f

B((, z)

J

B((, zo)

as z

Equation (4) becomes

= Bb(l)(Zo) — urn f B((,zo).

Further Results

344

FIGURE 24.1

To prove (2), it therefore suffices to show

urn f

2

—C.)

E

By a translation, we may assume that z0 is the origin. We may also pull back the resulting integral via the change of scale map ( — e( for = 1. The inverse image of the set under this change of scale converges to half of a unit sphere in as —f 0. Since the Bochner—Martinelli kernel is also invariant under the unitary group in C". we may assume this half sphere is the set

{( =

(,,);

CI

= I and Im(1 > 0}.

Therefore, we must show

f b(() =

(5)

345

Kernels for strictly pseudoconvex boundaries

where b(() = B((,0). To see (5), let S be the corresponding half of the unit we have sphere with Im(j < 0. In view of (I) with D as the unit ball in

1= f b(()+ f b((). (ES

is a diffeomorphism from Moreover, the conjugation map ( C(() which changes the orientation by a factor of (—1 )fl. By an easy S to the unit sphere. This uses the fact that on the computation, C'b (— We obtain sphere, 0 = d( . d(. + ( =(

fh= b.

This together with the previous equation implies that (5) holds and so the proof of (2) is complete.

24.2

Kernels for strictly pseudoconvex boundaries

For a compact convex boundary M = -, we use the complex gradient of a suitable defining function p as the generating function ?i for the kernels L, Lt, H, R, etc. The key properties possessed by u are the following.

(I) The function u((,z) is holomorphic in z e D for each fixed ( E M. (2) There are positive constants 6, and C such that if ( belongs to a 6neighborhood of and if K — zI <€, then

2Re{u((, z) ((— z)} p(() — p(z) + C(I( (3)

—

z12).

If z is in a 8-neighborhood of D, (is in a b-neighborhood of M, and if K

—

€,

then

AU the global results in Chapters 20 and 21 hold for any domain D with boundary M in which there is a generating function u that satisfies the above three properties.

In [Hell, Henkin constructs a generating function u for a bounded, strictly pseudoconvex domain, such that u satisfies properties (1), a modified version of (2), and (3). In this section, we outline his construction. This construction

Further Results

346

is not necessary for the local solution for the tangential Cauchy—Riernann complex because a strictly pseudoconvex hypersurface is locally biholomorphic to a strictly convex hypersurface. Henkin's construction of the kernel generating function requires the solution of the Cauchy—Riemann complex (with, say, Hörmander's £2-techniques [Ho]) on a bounded strictly pseudoconvex domain. This is somewhat unsatisfying since one of the applications of the generating function is the construction of integral kernel solutions to the Cauchy—Riemann complex on a bounded strictly pseudoconvex domain (i.e., the analogue of Theorem 1 in Section 20.3). Range has developed a self-contained (but more complicated) integral kernel approach to the problem of finding kernel generating functions and hence the solution to the Cauchy—Riemann complex and other results in several complex variables. We refer the reader to [Rani for his approach. Henkin's approach is to first locally construct the generating function u so that (1) and (2) hold (locally). He then modifies u by solving the appropriate 0-problem (globally) so that (1) and (3) hold. For the local construction, we {p(z)
((j-z3)((k

p(z) =

-

(4)

j.k—_1

where

Op(()

1

(ç--z3)((k—zk). sk

Defineü=(ü1 u3(çz)=

Op(c)

oc.

02p(()

1

((k—zk).

k=I

((— z) = Q((, z). Since D is strictly pseudoconvex, we may assume that p is strictly plurisubharmonic and so the complex hessian of p at each point in a neighborhood of M is positive definite. Equation (4) now shows that property (2) holds for ü. Clearly, property (I) also holds for ii. Before we modify ü, we first modify Q((, z) so that the resulting function is holomorphic in z and nonvanishing for (in a 6-neighborhood of M and for z e. We shall take the 5-neighborhood in a 6-neighborhood of D with ( — of M to be a set of the form {( p(()I <5} and the 5-neighborhood of S and if z E D5 with < 6). If D to be the set D5 = {z E then the inequality in (2) implies K — zI Note that

z)

347

Kernels for strictly pseudoconvex boundaries

Choosing 6 small relative to means that we can arrange z) > 0 for z) <6 and z E D6 with K — zI e/2. On this set, a branch of log is well defined and holomorphic in z. x Choose a smooth cutoff function x '—÷ CT1 such that 1

1

if 6, consider the 1-form

For fixed ( with

f(z)—f —

otherwise.

10

has smooth coefficients and it is a-closed on The form We can use the a-theory fora bounded strictly pseudoconvex domain to find a solution to the equation on D6. The function is smooth for = < 6. Moreover, for each fixed in this set, the function and z — is holomorphic in z ED6 with ( — e/2.

<6, define

For

Qf

—

f

—

1

if z ED6 with if z D6 with

K — zi K

—

function Q((, z) is well defined and holomorphic in z for z E 1)6 and 6. Also, Q((, z) 0 for K — zI e/2. Now the idea is to perform a division to find a function u : CT1 x C1 so that u((, z) is holomorphic in z E D6 ( for 6) and so that u(ç, z) (ç — z) = z). The basic idea is the following. The function Q((, z) vanishes on the diagonal {(z, z); z E Db}. For fixed the functions — Zn) locally generate the sheaf of holomorphic functions (in (ct — z1 ),. .. , z) which vanish at z = over the ring of germs of holomorphic functions at (. The

Since D. is a domain of holomorphy, these functions also globally generate the space of global sections of this ideal (this follows from the theory of coherent

analytic sheaves; see Chapter 24 in Hörmander's book [Ho]). So there exist functions z),..., z) which are holomorphic in z E D6 such that — z3) = Q(ç,z). Care must be taken to ensure that the u3((, z) depend smoothly on ( as well as z for < 6. We refer the reader to [Hel] for details. By the above construction, u satisfies properties (1) and (3). It is not clear that property (2) is satisfied. However for K — zi €/2, we have

u((.z)

.

- z) =

.

((- z)

and we know that Ii satisfies property (2). From the definitions of our kernels (see Section 20.1), it follows that if z) is a nonvanishing smooth function

348

Further Results

of ( and z, then E(cii) = E(ui) and E(ciii,c112) =

and so forth. So for the parts of the proofs of the theorems in Chapters 20 and 21 for the strictly pseudoconvex case that require the estimate in (2), we can replace u by ft in the appropriate kernels and then the proofs go through without further changes.

24.3

Further estimates on the solution to ÔM

R((, z) is locally integrable in E M (locally uniformly in z), there is some gain in the regularity of the global solution for the tangential Cauchy— Riemann complex given by the R kernel. From the dominated convergence Since

theorem, it follows that if f is a compactly supported form with £°° coefficients, then R(f) is a form with continuous coefficients. More generally, Henkin

[He3] has shown that if f is a compactly supported form with coefficients

in JY with I < p < 2n, then R(f) is a form with coefficients in C' with 1/r = (l/p)—(l/2n). If p = 1, then Henkin obtains r

(2n + 2)/(2n + 1)—f,

for any > 0. In addition, R(f) has C'°-coefficients provided the coefficients of f are compactly supported and belong to for any c> 0. Estimates on the local solution for the tangential Cauchy—Riemann complex

are more difficult than for the global solution. Indeed, there does not yet appear to be a local kernel solution for the tangential Cauchy—Riemann complex that exhibits a gain in regularity (maybe none exists). The solution given in Chapter 22 does not even preserve regularity due to the presence of the term E121([Ow.x]°" A 1). The best result available in this context is a recent result of Shaw [Sh2] who shows that a modification of the solution operator given in Chapter 22 is continuous in for 1

, given in Chapter 22 does not necessarily have continuous boundary values on 8WA due again to the presence of the term A f). In Section 5 of [He3], Henkin modifies this solution to one that is continuous up to 9WA provided f is a continuous (p, q)form with I q < n — 3. This solution operator (different than Shaw's) is also continuous in the on wX.

24.4

Weakly convex boundaries

The full strength of properties (2) and (3) in Section 24.2 for the generating function u or ü is not needed for some of the results in Chapters 20, 21, and 22 Many of these results only require property (1) in Section 24.2 (i.e., that u((, z)

Solvability of the tangential Cauchy—Riemann complex in other geometries

is holomorphic in z E

349

for (in M) and the following weaker version of

properties (2) and (3).

The function u((, z).((—z) is nonvanishing for (çz) E x Moreover, V is a neighborhood of Mx M in if ((o, zo) E V with u((o, zo) ((o — zo) = 0, then the real derivatives and z '—' u((o,z).((o—z) have of the maps respectively. maximal rank at the points (= (o and z = For a bounded convex domain that is not strictly convex, a defining function can be found so that its complex gradient satisfies this weaker property. Harvey and Polking show in [HP] that integral kernels can be constructed for the solution

of the 0-problem on domains that have generating functions that satisfy the above weaker condition. Their approach is somewhat different than Henkin's. Harvey and Policing take principal value limits across the singular sets of these kernels (i.e., the set {((, z) V; u((, z) ((— z) = 0}). As a result, they obtain new integral kernel solutions for the equation &u = on convex domains. These new solutions can be applied to more subtle problems. For example,

f

if I

smooth on D (but not necessarily up to M), then they provide an integral kernel solution u (in certain bidegrees) which is smooth on D. If f has compact support then they also obtain a solution u with compact support is

(in certain bidegrees).

The boundary value results for the L, Lt, H, and Ht kernels given in Theorems 1 and 2 in Section 20.4 also only require that the generating function u satisfy the above weaker condition. So these theorems (and the result concerning a fundamental solution for 0M given in Theorem 1 of Section 21.1) also hold for boundaries of weakly convex domains. In addition, the boundary values of one normal derivative of Af)JD- and of Af)ID-I. exist on M (Theorems I and 2 in Section 20.4 only discuss the boundary values of tangential derivatives). These and related results can be found in Sections 8 and 9 of [HP].

24.5

Solvability of the tangential Cauchy—Riemann complex in other geometries

The goal of much of Part IV is the discussion of the local and global solvability of the tangential Cauchy—Riemann complex on a strictly pseudoconvex hypersurface in In this section, we briefly discuss the local solvability of the tangential Cauchy—Riemann complex in other geometries. We fix a point zO in

a smooth real hypersurface M in C" and fix an integer q with 1 5q Sn— I. We ask the following question. What conditions on the Levi form of M at zO are needed to ensure that the equation = f can be solved near zo where f

350

Further Results

is a given smooth OM-closed (p, q)-form on M near Zqj? The answer is that 1k! must satisfy the Y(q) condition of J. J. Kohn at the point z0. The hypersurface M is said to satisfy condition Y(q) at the point z0 if the Levi form of M at ZO has either max{n — q, q + l} eigenvalues of the same sign or min{n — q, q + l} pairs of eigenvalues of opposite sign (i.e., min{n — q, q + 1 } positive eigenvalues and min{n q, q + 1 } negative eigerivalues). If M is strictly pseudoconvex at z0, then M satisfies condition Y(q) at z0 for all 1 q n — 2. More generally, suppose the Levi form of M at 21J has r-positive and s-negative eigenvalues with r + s = n — 1, then M satisfies condition Y(q) at z0 for each 0 q ii — s. Folland and Kohn use £2 techniques in their book except q = r and q

[FK] to show that if M satisfies condition Y(q) at every point zo in M, then the equation = is globally solvable where f is any smooth OM-closed

f

(p, q)-form. Local solvability under condition Y(q) was established by Andreotti and Hill [AnHi2J. We shall outline an integral kernel approach to Andreotti and Hill's result. For details, see [BS}.

Since there are several cases to consider for the condition Y(q), we shall discuss the case where the Levi form of M at ZO has (q+ 1)-positive eigenvalues with q + I n — q. The other cases of condition Y(q) are similar to this one.

We assume that zO is the origin and M is graphed over its tangent space at the origin as M = {z E p(z) = 0) where

p(z) =Imz1 Here, we have diagonalized the Levi form of M at the origin and the numbers P2,.. . , p7, are its eigenvalues. By replacing p with p + Cp2 as in the proof of Theorem i in Section 10.3, we may assume that the complex hessian of p at the origin is positive definite in the z1 -direction. Since the Levi form of M at the origin has (q + 1)-positive eigenvalues, we may assume that P2,... are positive. After a change in scale, we obtain n

+

p(z) = Imz1 +

+ O(1z13)

3=1

where

(I)

In the strictly pseudoconvex case, q + 2 n and the function used to generate the kernels in Chapters 20 and 21 is given by the complex gradient of p. Here, '—* x ui,) : be given by we let u' = . .

,

if

if

of the tangential Cauchy—Riemann complex in other geometries

351

A Taylor expansion argument together with the estimate in (1) imply that there is an open set U in containing the origin and a constant C> 0 such that

2Re{n'((,z).((—z)} p(() —p(z)+C(I(—z12) for (,z E U. Let u2((, z) = u'(z,

Reversing the roles of ( and z in the

above inequality yields

2Re{u2((,z)

— z)}

p(()

— p(z)

—

C(I( — zt2)

for (,z EU. We form the kernels L = E(u'), Lt E(u2), and R = E(u',u2) as in Chapters 20 and 21. The above estimates imply that L((, z) is smooth

for ( E Mn U, z E Un D and Lt((,z) is smooth for ( E Mn U and z E UflDt where D = {z E = {z E C'2; p(z) > 0}. p(z) <0} and Moreover, R((, z) has only diagonal singularities on M x M. The proof of Theorem 2 in Section 20.4 can be repeated using the above estimates to show

flU) then L(f) is smooth on the set Un D and Lt(f) is

that if f E

smooth on the set U fl D+. The proof of Theorem 1 in Section 21.2 is therefore valid in this context and we obtain

0R+R0OM +(Lt)+ -

L

=I

as operators on (M n U). In the strictly pseudoconvex case, we show (Lemma 3 in Section 20.3) that the L kernel acts nontrivially only on (p, 0)-forms and the Lt kernel acts nontrivially only on (p, n — 1)-forms. This together with (2) means that the R kernel is a fundamental solution for the tangential Cauchy—Riemann complex (acting on for 1 q n— 2. in our setting where M satisfies condition Y(q) at the origin, we claim that L(f) and Lt(f) vanish if f E flU) and so the R kernel defined by our new support function u1 is a fundamental solution to the tangential Cauchy— Riemann complex (acting on n U)). To see this, we examine the (Ou' d(( — z))'2' part of the kernel Let u(() = We have n—I

d(( -

d((-z)+

=

d((2 -

A d((3 -

j=q+3

J

(n_l) k

A

Further Results

352

The only contributing terms to the sum on the right occur when k n — q — 2, for otherwise there will be a repeated wedge product of d((3 — z3) for some j with q±3 j n. Together with the fact that u only depends on we see that in d( is at least n—i —(n—q—2) = q+ 1. So the degree of (Ou' flU) with s n—q— 1, then L(f) = 0. Since q n—q— 1 (by if f E assumption), we have L(f) = 0 for f E fl U), as desired. In a similar fl U).It follows that I? is a manner, we can show Lt(f) = 0 for f E fundamental solution for 0M on flU), i.e., f = OM{R(f)} +R(9Mf). To modify the fundamental solution for the tangential Cauchy—Riemann complex to handle the local problem, we use a procedure similar to the one in Chapter 22. In that chapter, we consider a local neighborhood basis of open sets in M about the origin obtained by slicing M with the half space Rez1 > —A for A > 0. Since M is strictly convex (in Chapter 22), these open sets shrink down to the origin as A '—+ 0.

in this section, M is only strictly convex in the (zi,...

,

zq+2)-directions.

Therefore, we shall obtain a local neighborhood basis of open sets by intersecting M with a trough that is flat in the . , )-directions and that "bends up" More precisely, let in the (Zq+3,. , .

.

WA =

EM; Imz1 > —A+2

1z312}.

j=q+3

The choice of the number 2 is motivated by the estimate (1) on the eigenvalues which ensures that the above trough bends up faster than M in 11q+3' .. , the directions (zq+3, ... , zn). In fact, combining the defining equation for M and the defining equation for WA, we obtain q+2

forzEwA. j=1

The estimate in (1) together with this inequality imply that the diameter of W), 0. From now on, we is proportional to VX. So WA shrinks to the origin as A restrict A so that WA is contained in U. Let

r(z) = Imz1

—2

j=q+3

The defining equation for WA is given by {z E M; r(z) > i—' C'3 by C" x

—A}.

Define

As in Chapter 22, we form the kernels E13, E23, and E123. As in the proof of part (a) of Lemma 1 in Section 20.3, the (weak) convexity of r implies that if

Solvability of the tangential Cauchy-Riemann complex in other geometries

zE

and ( A

then u3((, z) ((— z) A f),

0. So if f E

353

then

A 1) are smooth forms

On W.,.

The same arguments used for the L kernel above allow us to show that the degree of E13((, z) in d( is at least q. Therefore, E13 acts nontrivially only

on currents of bidegree (p, s) with s

n—q—

1.

Since q n — q —

1,

E13([OwAI°'1 A f) vanishes for f E As with the strictly pseudoconvex case, the term E23([Ow,j°" A 1) does not vanish purely from bidegree considerations. However, if f is a then an approximation argument similar to that at smooth (p, q)-form on

the end of Chapter 22 can be carried out to show that E23([aw.x]°" A f) vanishes. In this case, the kernel E23((, z) is (formally) holomorphic only in the So the approximation argument must be carried out in variables (it... treated as parameters. We refer the reader to these variables with (q+3, . . the end of [BS] for details. A f) both vanish if f is a smooth, — Since E13([OwA]°" A f) and the procedure in Chapter 22 can be carried out in q)-form on aM-closed is given by (3) in Chapter 22. this context so that the solution to OMU = on In the strict pseudoconvex case, we showed in Chapter 23 (with Lewy's example) that the tangential Cauchy—Riemann complex is not locally solvable at top degree (q = n — 1). More generally, if the Levi form of a hypersurface at a point zo has p-positive eigenvalues and q-negative eigenvalues, then the .

,

f

tangential Cauchy—Riemann complex is not locally solvable in degrees p and q. This is a result of Andreotti, Fredricks, and Nacinovich [AFN}. Generalizations of the results in this section to manifolds of higher codimension have been obtained by Airapetyan and Henkin [AH1], [AH2].

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K. Yosida, Functional Analysis. Springer-Verlag, NY, 1980.

Notation

The following is a partial list of notation. The page number refers to the page where the notation is first encountered or defined. space of smooth functions on — p. 2. the space of compactly supported smooth functions on — p. 2. the space of real analytic functions on — p. 4. the space of holomorphic functions on Il — p. 6. the space of Holder continuous functions with exponent — p. 215. the norm on the space — p. 216. T(X) the tangent bundle to the manifold X — p. 24. Tc (M) the complexified tangent bundle to M p. 40. 11(M) the complex tangent bundle to a CR manifold M p. 97. 11C (M) the complexification of H(M) — p. 101. X(M) the totally real tangent bundle to a CR manifold M — p. 98. the cotangent bundle of X — p. 26. (M) the complexified cotangent bundle of it'! — p. 40. the bundle of r-forms on Al — p. 27. (X) the bundle of complexified r-forms on X — p. 40. J a complex structure map — pp. 7, 41. the induced complex structure map on forms — p. 42. VLO (resp. V°") the +i (resp. —i) eigenspace of J on the vector space V — the

p.

43. the

bundle of (p,q)-forms on Al — p. 46.

see p. 123. the space of smooth r-forms on M — p. 27. 17(M) the space of smooth r-forms with compact support on M — p. 79. the space of smooth (p, q)-forms on a complex manifold M — p. 46. the space of elements of with compact support — p. 49. the space of smooth (p, q)-forms on a CR manifold Al — p. 124. the space of elements of with compact support — p. 124. the space of smooth sections of — p. 124.

359

Notation

the

space of smooth sections of

with compact support — pp. 124,

128. D'(Il) the space of distributions on Il — p. 62.

£'

the space of distributions with compact support on the

—

p. 62.

space of currents of degree r on — p. 81. the space of currents of degree r with compact support on —

p. 81.

the space of currents of bidegree (p, q) on — p. 83. the space of currents of bidegree (p, q) on a CR manifold M — p. 129. F defined on vectors (or currents) — p. 8. Ft the pull back map via F defined on forms (or currents) — p. 11. DF the matrix which represents the derivative of F — p. 7. d or dM the exterior derivative operator — p. 28. d* the £2-adjoint of the exterior derivative on RN — p. 16. o the Cauchy—Riemann operator — p. 47. 0* the £2-adjoint of 0 on CM — p. 49. °M the tangential Cauchy—Riemann operator (either extrinsic or intrinsic) — p. 124.

the tangential piece of the form f along M —

ftM dci

a

d/.SM (

,

)c2

p. 124.

volume form — p. 36. surface measure on M — p. 63. pairing between forms and vectors at the point p — p. 9. pairing between distributions and functions on — p. 62. pairing

between currents and forms on — p. 80.

the intrinsic Levi form at p — p. 156. the extrinsic Levi form at p — p. 160. the convex hull of the image of — p. 200. F1
T the Hubert transform in Part III — p. 214. If the Fourier transform of f p. 231.

If

the modified Fourier transform of f — p. 232. WF the hypoanalytic wave front set — p. 238. [M] the current given by integration over the manifold M p. 81. the current given by integration over the diaognal — p. 83. K' the adjoint of the kernel K — p. 267. Kt the transpose of the kernel K — p. 269. a particular type of kernel defined on p. 295. E1 or E(u2',... , B the Bochner—Martinelli kernel — p. 284. L the Leray kernel — p. 301.

H the Henkin kernel — p. 301. R the Romanov kernel — p. 318. seep. 307. see p.307.

Index

Airapetyan, A., 353 almost complex structure, 58 analytic disc, 206—228

canonical local basis, 109 Cauchy kernel, 216, 217 on a slice, 282

boundary of, 206

Cauchy—Riemann operator, 47

theorem, 207, 221—228

Chang, C. H., 229, 256 change of variables formula for integrals, 30, 33 Cirka, E. M., 342 complex, 134, 135, 263 complex structure on a more general vector space, 41 on C11,7 complex tangent space, 97 complexification (of a vector space), 39 complexified tangent bundle, 40 conjugation, 39 contraction (of a form by a vector), 15, 49 convolution of distributions, 66 of functions, 3 coordinate chart, 17 patch, 17 CR codimension, 98, 121 distribution, 140, 255, 256 equivalence, 154 function, 140 manifold, 121

Andreotti, A., 169, 350, 353 approximation by entire functions, 191—197 by polynomials, 5

Baeouendi, M. S., 110, 145, 183, 184, 191, 229, 256—258, 260—262

bidegree of a current, 83 of a differential form, 46 bidimension (of a current), 83, 134 Bishop, E., 214 Bishop's equation, 215, 220 Bloom, T., 105, 181 Bochner, S., 264, 284, 292 Bochner—Martinelli kernel, 284, 285, 300 boundary values, 286 as a fundamental solution for 8, 285 jump formula, 286 principal value limits of, 342 Bochner's extension theorem, 292 Boggess, A., 200, 258, 260, 261, 350 boundary point, 23, 32

Caley transform, 337

map, 149—155

structure (same as CR manifold), 121 submanifold, 99 current

361

362

Index

definition of, 80 operations with, 84—94

cutoff function, 3

D'Angelo, J. P., 180 degree

ofa current, 81 ofa form, 12 delta function, 62, 68 diagonal, 83, 266 differentiable function (on a manifold), 18 differential form on a manifold, 27, 46

on RN 9, 12 dimension (of a current), 80, 81 distribution definition, 62 operations with, 65—70

support of, 63 divergence theorem, 36 Dwilewicz, R.. 258

extension theorems for CR functions, 141, 147, 198—202, 239—250, 292, 315, 333

exterior derivative of a current, 85 exterior derivative of a form on a manifold, 28, 41, 47 on RN, 9, 12

for the Laplacian, 76 for the tangential Cauchy—Riemann operator, 314, 318

generic, 102, 187 Graham, I., 105, 181 graph of a function, 20, 267 Grauert, H., 323 Green's formula, 37

Harris, G. A., 188 Harvey, R., 264, 286, 343, 349 Heisenberg group, 113. 336 Henkin, G. M., 264, 294, 304, 315, 318, 327. 338, 345, 348, 353 Henkin's kernels, 295 as a fundamental solution for 0M. 314 boundary values of, 304 definition of, 301 Hubert transform, 214, 217 Hill. C. D., 169, 187, 260, 350 Holder continuous, 215 holomorphic function. 6, 19 map, 7 tangent space (same as complex tangent space), 97 homotopy formula, 272 Hbrmancler, L., 1, 59. 181, 210, 332, 345. 347

Federer, H., I finite type, 181 Folland, G. 8., 1, 57, 264, 335, 350 formal identity (for Henkin's kernels), 297 Fourier

Hdrmander number, 181, 257, 258 hypersurface, 20 hypoanalytic wave front set, 238, 253 hypoellipticity, 260

inversion formula, 235, 25 1—254

transform (modified), 231, 232, 253 Fréchet space, 62 Fredricks, G., 353 Freeman, M., 186 Frobenius theorem analytic version, 56—58

smooth version, 52, 159, 186 fundamental solution for 9, 274, 281—285 for a partial differential operator, 75 for the Cauchy—Riemann operator on C, 75 for the exterior derivative, 274, 278—280

identity theorem, 6, 142, 148 imbeddability of CR manifolds, 121. 169—178

inner product, 44 Euclidean, 9 16, 49 integral curve, 54 integration of differential forms, 30—32, 41

involutive, 52, 101

Jacobowitz, H.. 145, 172. 256, 261, 262 John, F., I , 57

Index

363

kernel

adjoint of, 267 definition of, 266 of convolution type, 270 on a ray, 278 transpose of, 269 Krantz, S., 1, 316 Kuranishi, M., 187

induced, 30 preserving map, 29 oriented, 29 coordinate chart, 30 manifold, 30 partition of unity, 3 Pincuk, S. I., 261 Pitts, J., 260 Poisson's integral formula, 216 polar (of a cone), 239 Polking, J. C., 130, 264, 286, 343, 349 polynomial growth, 256 projective space, 18 pseudoconvex, 165 strictly, 164, 187, 346 pull back of a current, 91, 92

Lawson, B., 343 Leray kernel, 301 Levi flat, 158 Levi form coordinate formula for, 160, 162, 198 definition of, 156, 160 Levi null set, 185 Lewy, H., 198, 261, 264, 334 Lie algebra bracket. 14, 179 group, 112, 183 subalgebra, 183 Lieb, 1., 323 local defining system, 20

ofa form, 11, 27, 48 pure term, 106, push forward of a current, 87 of a vector, 8, 25

Maire, H.-.M., 262 manifold

quadratic form, 111 quadric submanifold, 111, 210

complex, ig CR, 121

point, 23, 32 real analytic, 17 smooth, 17 minimal, 257 moment condition, 293 Nacinovich, M., 353 Nagel, A., 258, 259 Newlander, A., 59, 121 Newlander—Nirenberg theorem, 59, 121, 159, 186 abstract version, 59 imbedded version, 59 Nirenberg, L., 59, 121, 165, 169, 172 Nirenberg's nonimbeddable example, 172—178

normal form, 105, 109 Bloom—Graham. 181

for codimension 2 quadrics, 114

Ramirez de Arellano, E., 294 Range, M., 346 real analytic function, 4, 19 manifold, 17 rigid, 110, 229 Romanov kernel, 318 Rothschild, L. P., 110, 183, 184, 229, 256—258, 262

Rosay, J.-P.. 262 Rossi, H., i

Schwartz, L., I second fundamental form, 166 semirigid, 184 Shaw, M.-C., 262, 348, 350 SjOstrand, J., 229 solvability

ofä(globally), 315, 316, 318 of Ô (locally), 327, 338 of 0 on a strictly convex domain, 303

orientation, 29

spherical kernel, 278—280

Index

364

Spivak, M., I Stein, E. M., 258, 259 Stokes' theorem, 33 subbundle, 24, 40, 57 submanifold, 19

with boundary, 22

Taiani, G., 187, 260 tangent bundle, 24 space, 24 tangential Cauchy—Riemann complex,

of a kernel, 266, 271 of a point, 179, 257 uniqueness theorems, 141, 142. 198—199, 200—201. 202

vector, 7, 23 bundle, 25 field, 8, 24, 40 volume form, 36

122—139

equivalence of extrinsic and intrinsic

definitions, 135 extrinsic definition, 124 intrinsic definition. 131 tangential vector field, 304 Tomassini, G., 141 totally real, 98, 99 transversal Lie Group action, 183 Trepreau, J. M., 258 Treves, E, 110, 145, 172, 183, 191, 229,

Wainger, S., 259 weak topology, 64, 68 Wedge, 196

wedge product of currents, 84 of forms, 10 Weierstrass theorem, 5, 191, 192 weight. 180

Wells, R. 0. Jr, 25, 130, 188 Whitney extension theorem, 71, 72

256, 260—262

tube like, 261 Tumanov, A. E., 257 type

Yosida, K., I Y(q) (condition), 350

'152 ISIS,, I?