Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
Subseries: Fondazione C. I. M. E., Firenze Advisor: Roberto Conti
1684
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
C. A. Berenstein R E Ebenfelt S.G. Gindikin S. Helgason A.E. Tumanov
Integral Geometry, Radon Transforms and Complex Analysis Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Venice, Italy, June 3-12, 1996 Editors: E. Casadio Tarabusi, M. A. Picardello, G. Zampieri
Fondazione
C.I.M.E.
Springer
Authors
Editors
Carlos A. Berenstein Institute for Systems Research 221 A. V. Williams Building University of Maryland College Park, MD 20742-0001, USA
Enrico Casadio Tarabusi Dipartimento di Matematica "G. Castelnuovo" Universith di Roma "La Sapienza" Piazzale Aldo Moro, 2 00185 Roma, Italy
Peter F. Ebenfelt Department of Mathematics Royal Institute of Technology 100 44 Stockholm, Sweden Simon Gindikin Department of Mathematics Hill Center Rutgers University New Brunswick, NJ 08903-2101, USA Sigurdur Helgason Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA
Massimo A. Picardello Dipartimento di Matematica Universith di Roma "Tor Vergata" Via della Ricerca Scientifica 00133 Roma, Italy Giuseppe Zampieri Dipartimento di Matematica Pura ed Applicata Universit'~ di Padova Via Belzoni, 7 1-35131 Padova, Italy
Alexander Tumanov Department of Mathematics University of Illinois 1409 West Green Street Urbana-Champaign, IL 61801-2943, USA
Cataloging-in-Publication Data applied for Die Deutsche Bibtiothek - CIP-Einheitsaufnahme Integral geometry, radon transforms and complex analysis : held in Venezia, Italy, June 3-12. 1996 / C. A. Berenstein ... Ed.: E. Casadio Tarabusi ... - Berlin; Heidelberg; New York; Barcelona: Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1998 (Lectures given at the ...session of the Centro lnternazionale Matematico Estivo (CIME) ... ; 1996,1) (Lecture notes in mathematics; vol. 1684: Subseries; Fondazione CIME) ISBN 3-540-64207-2 Centro Internationale Matematico Estivo
: Lectures given at the ... session of the Centro lnternationale Matematico Estivo (CIME) ... - Berlin; Heidelberg; New York; London; Paris; Tokyo; Hong Kong: Springer Friiher Schriftenreihe. - FriJher angezeigt u. d. T.: Centro lnternationale Matematico Estivo: Proceedings of the ... session of the Centro l nternationale Matematico Estivo (C1ME) 1996,1. Integral geometry, radon transforms and complex analysis. - 1998 Mathematics Subject Classification (1991): 43-06, 44-06, 32-06 ISSN 0075- 8434 ISBN 3-540-64207-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10649783 46/3143-543210 - Printed on acid-free paper
PREFACE
This book contains the notes of five short courses delivered at the Italian Canfro Internazionale Matematico Estivo (CIME) session Integral Geometry, Radon Transforms and Complex Analysis held at Ca' Dolfin in Venice (Italy) in June 1996. Three of the courses (namely those by: Carlos A. Berenstein of the University of Maryland at College Park; Sigurdur Helgason of the Massachusetts Institute of Technology; and Simon G. Gindikin of Rutgers University) dealt with various aspects of integral geometry, with a common emphasis on several kinds of Radon transforms, their properties and applications. The lectures by C. A. B e r e n s t e i n , Radon transforms, wavelets, and applications, explain the definition and properties of the classical Radon transform on the two-dimensional Euclidean space, with particular stress on localization and inversion, which can be achieved by the recent tool of wavelets. Interesting applications to Electrical Impedance Tomography (EIT) are also illustrated. The lectures by S. Helgason, Radon transforms and wave equations~ give an account of Radon transforms on Euclidean and symmetric spaces, focusing attention onto the Huygens principle and the solution of the wave equation in these environments. The lectures by S. G. Gindikin, Real integral geometry and complex analysis, give an account of the deep connection between the two main themes of this CIME session, covering several variations of the Radon transform (RT): the projective RT; RT's taken over hyperplanes of codimension higher than 1; and RT's over spheres. An important and unifying tool is the ~" operator of Gel'fand-Graev-Shapiro, used to explain analogies between inversion formulas for the various RT's. This approach goes hand-in-hand with ~-cohomotogy and hyperfunctions, typical subjects in the field of complex a~alysis. In related areas, the other two courses (namely those by: Alexander E. Tumanov of the University of Illinois at Urbana-Champaign; Peter F. Ebenfelt of the Royal Institute of Technology at Stockholm) share stress on CR manifolds and related problems. The lectures by A. E. T u m a n o v , Analytic discs and the extendibility of CR functions, provide an introduction to CR structures and deal in particular with the problem of characterizing those submanifolds of C N whose CR functions are wedge-extendible. This property turns out to be equivalent to the absence of proper submanifolds which carry the stone CR structure. (The technique of the proof consists in an infinitesimal deformation of analytic discs attached to CR submanifolds.) The lectures by P. F. E b e n f e l t , Holomorphic mappings between real analytic aubmanifolds in complex space, deal with algebralcity of locally invertible holomorphic mappings. Along with classical results, new criteria are introduced in terms of the behavior of these mappings on a real-analytic CR submanifold which is generic, minimal, and holomorphically non-degenerate in a suitable sense. To this end a fundamental tool is afforded by the so-called Segre sets.
VI We wish to express our appreciation to the authors of these notes, and to thank all the numerous participants of this CIME session for creating a lively and stimulating atmosphere. We are particularly grateful to those who contributed to the success of the session by delivering very inspiring talks. Enrico C A S A D I O T A R A B U S I Massimo A. P I C A R D E L L O Giuseppe Z A M P I E R I
TABLE OF CONTENTS
BERENSTEIN, C. A. Radon Transforms, Wavelets, and Applications Holomorphic Mappings Between Real Analytic Submanifolds in Complex Space
35
GINDIKIN, S. G.
Real Integral Geometry and Complex Analysis
70
HELGASON, S.
Radon Transforms and Wave Equations
99
TUMANOV, A. E.
Analytic Discs and The Extendibility of CR Functions
123
EBENFELT, P. F.
Radon transforms, wavelets, and applications Carlos Berenstein
We present here the informal notes of four lectures 1 given at Cs Dolfin, Venice, under the auspices of CIME. They reflect the research of the author, his collaborators, and many other people in different applications of integral geometry. This is a vast and very active area of mathematics, and we try to show it has many diverse and sometimes unexpected applications, for that reason it would impossible to be complete in the references. Nevertheless, we hope that every work relevant to these lectures, however indirectly, will either be explicitly found in the bibliography at the end or at least in the reference lists of the referenced items. I apologize in advance for any shortcomings in this respect. The audience of the lectures was composed predominantly of graduate students of universities across Italy and elsewhere in Europe, for that reason, the emphasis is not so much in rigor but in creating an understanding of the subject, good enough to be aware of its manifold applications. There are several very good general references, the most accesible to students is, in my view, [Hell. For deeper analysis of the Radon transform the reader is suggested to look in [He2] and [He3]. For a very clear explanation of the numerical algorithms of the (codimension one) Radon transform in R 2 and R 3, see [Na] and [KS]. There have also been many recent conferences on the subject of these lectures, for a glimpse into them we suggest [GG] and [GM]. Finally, I would like to thank the organizers, Enrico Casadio Tarabusi, Massimo Picardello, and Giuseppe Zampieri, for their kindness in inviting me and for the effort they exerted on the organization of this CIME session. I am also grateful to David Walnut for suggestions that improved noticeably these notes. 1. T o m o g r a p h i c i m a g i n g of s p a c e p l a s m a Space plasma is composed of electrically charged particles that are not uniformly distributed in space and are influenced by celestial bodies. The problem consists in determining the distribution function of the energy of these particles (or of their velocities) in a region of space. A typical measuring device will take discrete measurements (for instance, sample temperatures at different points in space) and then the astrophysicist will try to fit a "physically meaningful" function passing through these points. The procedure proposed in [ZCMB] is based on the idea that the measurements should directly determine the distribution function. We do it by exploiting the charged nature of the particles and using the Radon transform. (The recently launched Wind satellite carries a measuring device based on similar interaction principles and requires tomographic ideas for the processing of the data.) The advantage of the tomographic principles that we shall describe presently is that each measurement carries global information and seems to have certain noise reduction advantages over the pointwise measurements of temperatures, which is the 1These lectures reflect research of the author partially supported by the National Science Foundation.
|
](
Figure 1: Schematic detector. usual technology. We will describe everything in a two-dimensional setting, but the more realistic three-dimensional case can be handled similarly. The instrument we proposed in [ZCMB] is schematically the following. An electron enters into the instrument (a rectangular box in the figure below) through an opening located at the origin and is deflected by a constant magnetic field/~ perpendicular to the plane of the paper (see Figure 1). Under the Lorentz force, the electrons follow circular orbits and strike detectors located on the front-inside surface of the box (along the y axis). Those that strike a detector located at the point y have the property that
where m is the mass of the electron, e its charge, and B the magnitude of the magnetic field /~. In other words, all the electrons with the same first component v~ of their velocities strike the same detector located at the height y. The range of velocities over a segment of width a (width of the detector) is
Avx = (eB/2m)d (In terms of the length of the detector plate D in Figure 1 and the maximum velocity vm~ we have Avx = (d/D) - Vm~). If f(v~,vy) represents the electron velocity distribution, then the number dN of electrons counted by a detector in time dt is given by
dN = Anev~Avx i f(vx, vy)dvy, dr, --00
ne is electron density and A is the area of the entrance aperture. In other words, 1 dN f(v~, v~)dvy = AneAv~--~ --oo
so t h a t the count of hits provides the integral of f along a line vx = constant in the velocity plane. By rotating the detector or changing the orientation of the magnetic field we obtain the Radon transform of f . As a realistic example, consider a plasma of nominal electron density ne = 10 c m - 3 , velocity in the range Vmin to Vmax of 1.2 X 10 s to 3.0 • 109 cm s -1, average velocity = 6.5 • 108 cm s -1, and we assume a Gaussian distribution function
so t h a t dt
-
const, e x p , ,
2~2,]
with individual detector area and aperture of 0.04 c m 2 for a small instrument one gets t h a t the distribution function f varies from 1 to 10 -5 while d N / d t varies from 102 to 105s -1. The s t a n d a r d measurement methods make the a priori assumption t h a t f is the sum of a Gaussian centered at V and perturbed by adding a finite collection of Gaussians, often located in the region where f varies from 10 -4 to 10 -5, but the previously described instrument does not require any such assumption, on the other hand, experimentally one sees that such large variations, like from 1 to 10 -5 as in the example, are realistic. We shall see in Section 2 t h a t this is an embodiment of the Radon transform in R 2. The more realistic case of 3-d is handled by an instrument where there is a plane which contains the entrance aperture and a 2-d array of detectors in the plane (x, y). One shows t h a t at each detector location (x, y) one obtains an integral over a planar curve and that the addition of overall elements with the same x component leads to a 2-d plane integral of the density distribution so t h a t we have the Radon transform in R 3. (This is an observation we m a d e jointly with M. Shahshahani.) Before concluding this section, let us remark t h a t the large variations expected from the velocity density function f make the inversion of the Radon transform very ill-conditioned, even if f is assumed to be a smooth function. This is due to the continuity properties of the Radon transform and its inverse as seen in the next section. The remarkable point is that in medical applications, like CAT scans, the unknown density is naturally discontinuous along some curves but otherwise it has small local variations, and it is this reason the inversion problem is ultimately easier for medical applications.
Source
I []
Detector
Figure 2: Schematic CAT scanner. 2. T h e R a d o n T r a n s f o r m in R ~ Let w E S 1, w = (cos 0, sin 0), and take p E R. The equation x-w = p represents the line l which has (signed) distance p fi'om origin and is perpendicular to the direction 02.
For any reasonable function f (e.g., continuous of compact support), we can compute the line integral, with respect to Euclidean arc length ds, oo
f(x)ds = / f(x0 + tw• )dt
Rf(w,p) := / X,o2~p
(1)
--00
where x0 is a fixed point in l, i.e., satisfying the equation x0 - w = p, and w • = ( - sin 0, cos 0) is the rotate of w by ~/2. The map f ~-~ R f is called the Radon transform and R f is called the Radon transform of f. Clearly R f is a function defined on S 1 • R (that is, the family of all lines in R 2) with the obvious compatibility condition:
(R f)(-w, -p) = Rf(w, p).
(2)
There are several reasonable domains of definition for R such as LI(R~), $(R2), etc., but in many applications it is enough to consider functions which are of compact support, with singularities which are only jumps along reasonable curves, and otherwise smooth. This is obviously the transformation appearing in Section 1. The full 3-d instrument there corresponds to integration over planes in R 3, perpendicular to a unit vector w. A big source of interest of this transform lies in CAT (Computerized Axial Tomography) as a radiological tool where each planar section of a patient is scanned by X-rays as in Figure 2. In this particular case it can be seen that
I0
f
log ~ ~ / p d s Jl
(3)
where I0 is the radiation intensity at the source and I is the intensity measured at the detector. The attenuation is a consequence of traversing a tissue of density #. So the data collected from this X-ray scanning appears in the form of the Radon transform R # of the density #, computed for a finite collection of directions wt, w2,..., w~v (usually equally spaced) and a finite collection of lines, i.e., values Pl,P2,...,PM for each direction. This is called a "parallel beam" CT scanner. The configuration that it is now most used but we shall not discuss here is the "fan beam" CT scanner, we refer to [Na], [KS] for a discussion of the differences of these two cases, they really only appear at the implementation level of the inversion algorithms because only a limited amount of data can be obtained in the real world. Some easy properties of the Radon transform are obtained by observing that Rf can be written using distributions. In fact, if we introduce the unit density 5 ( p - x . w ) which is supported by the line x 9w = p, then
Rf(w, p) = f f(x)~(p - x. w)dx
(1')
R ~
with the usual abuse of language. It is also convenient to write
R~f(p) = Rf(w,p). extend Rf to (R2\{0}) x R,
using the fact that
nf(~, s) = ~ n s
,
(4)
Formula (1') can be used to is homogeneous of degree - 1 ; indeed, one defines
5(p-x.w)
(5)
One can therefore take derivatives of (1') with respect to the variables ~j(~ = (~1, ~2)) and obtain
~---~jRf(~,s)=
f f(x)~-~j~i(s-x.~)dx
(j = 1,2),
but
~
5 ( ~ - ~ . ~) = - : J ( s
- x . ~)
and ~
- x . ~) = ~ ' ( s - z . ~)
so that
0
--Rf(~, s) o~
f f(x)xjh'(s - x" ~)dx O f f(x)xjh(s- :c'~)dx 0 Os (R(xjf)(~, s)).
(6)
On the other hand the Radon transform of the derivative of f is:
= fff
fj(x)a(s-z.r
= ~j f f(x)5'(s - x . r = ~jff---~(RJ)(s).
(7)
In particular, for 02 02 A = Oz~ + Ox~' one obtains
Rr
= (~ + ~)
(RJ)(s).
(7')
When ~ is restricted to be an element co E S 2, we get, 02
( R A f )(co, s) = -~s2s2R f (co, s);
(S)
In other words, R intertwines A and ~0 2 when the arguments are restricted to S 1 • R. Another useful property is the following: P ~ ( f 9 g) = P ~ f | P~g,
(9)
where the symbol 9 on the left side of (9) denotes the convolution in R 2 and | denotes the convolution product in R. The easiest way to verify (9) is via the Fourier Slice Theorem, which we recall here: Let ~1 denote the Fourier transform of a function in R and f or F2 the Fourier transform of a function f in R 2. Then .T'l(P~f)('r) = f(Tco).
(10)
The proof is as follows, oo
Y'I(R~f)(T)
=
f e-2~it~l~f(t)dt --00
= f e-2'm"f( tco + scoX)dsdt, R 2
Letting now x = tco + sw • one has t = x- w and dtds = dx, the Lebesgue measure in R 2, in the previous equation we obtain
.T'I(/~/)(T)
= f e-2"(:"')'V(z)dz It.2
iO-~).
=
Recalling that in
R2
9F'2(f * g)(~) = ](~)~(~), we can easily prove (9). Indeed, from (10) we have
~ I ( P ~ ( f 9 g))(.)
= ~2(: 9 g)(.~) = / ( . ~ ) ~ ( . ~ ) = yl(P~/)(.)~:l(p~g)(~) = ~:l(p~f | p~g)(.)
and therefore, by the injectivity of 5vl, we get P ~ ( f 9 g) = P ~ f | Let us also note that if Ta denotes the translation by a, i.e., T , f ( x ) then R(7-,~f)(w,p) = Rif (x - a)](w,p) = P~f(p
= f(x
- a),
- w . a) = % . a R f ( p ) .
We now proceed to state some inversion formulas, which give different ways to recover f from R f . Fourier Inversion Formula: oo
(nj)O-)d~ 0
(11)
$1
The proof is clear, we begin with the Inversion Theorem for the Fourier transform. We have
f(x) = [ R2
let ( = 7-w and integrate in polar coordinates, to obtain oo
0
w6S i
We now apply the Fourier Slice Theorem to get oo
0
wES1
This inversion formula can be implemented numerically using the Fast Fourier Transform (EFT) (see [Na]). Quite often the points T w where the data J : l ( R ~ , f ) ( ~ - )
is known do not have a lattice structure. This causes problems for the F F T but we can use rebinning algorithms like [ST] to obviate this problem. To obtain another kind of inversion formula we observe the following:
fi P~f(s)g(s)ds= fi ; S(sw + tw• -oo
Let x --
sw + tw•
-oo
so that s -- x 9w, dx
-oo
= dtds, and
-r
therefore
1%2
i.e., the adjoint of P~ is the operator R~ defined by
We now consider for an
f
Rig(x) = 9(x. ~). arbitrary function g(w, s), having
the s y m m e t r y
(12) g(-w, -s) =
Rf(w,s)g(w,s)dwds= fsl dw fi R~f(s)g(w,s)ds
SlxR
-~
oo
(with the usual substitution,
x = sw + tw•
etc., we get)
= fd~ff(x)g(~,~.x)d2x $1
R 2
= /f(x)R#g(x)d2x.
(13)
l:t2
The operator R # defined by (13) is known by the name of "backprojection operator". Note, in this regard, t h a t g(w, s) is a function of "lines" and that R#g(x) is its integral over all lines passing through x. It is easy to prove the following useful property of the backprojection operator
(R#g) 9 f = R#(g | Rf),
(14)
where the convolution | in the second member clearly takes place in the second variable. This identity plays an important role in the numerical inversion of the Radon transform. Finally, we get to the following important result:
R # R f = 2 9 f.
(15)
Indeed,
R#Rf(x)
= SRf(w'w'x)dw S~
=
Sd,~zS f ( ( m - x ) m + s w • S1
--00
S~
By setting y = s~o• s =
lyl, dy = s&ods
0
we get
R#RS(x) = ~ S ~S(x § y)<~y R2
= which is exactly (15). Since one has that
2ST;--m~s(~,)y, 1
d
R2
2 ~1 = 1 (see e.g. [He2, page 134]), one deduces
m2(n#n/)(~) = ~](~) One can therefore conclude t h a t the inversion operator A is such t h a t
X(~) = so t h a t
'S e2~'~l~l(R#Rf)~(~)d~ = AR#Rf(x),
f(x) = ~
(16)
R2
which is sometimes called the backprojection inversion formula. Reorganizing the terms in the last formula one can rewrite it to obtain a more s t a n d a r d form, where the filtering is one dimensional.
10 F i l t e r e d b a c k p r o j e c t i o n inversion formula:
f = 1R#H(Rf)'
(17)
where
(Rf)'(w, s) = (~sP~f)(s) and H is the Hilbert transform defined by
Hg(s)= l-Tr? f(~t-)tdt' --00
where the last integral is understood in the sense of principal value. In other words,
1 (Rf)'(O,t_)dtdO f(~)- (2~)~L, f x.O-t 9
(18)
R
No introduction to the Radon inversion formula can be complete without at least mentioning the inversion formula due to Radon, which among other things, is akin to the inversion formula for the hyperbolic Radon transform due to ttelgason, which will be mentioned below. Consider for a fixed x E R 2, the average of Rf over all lines at a distance q > 0 from x, namely, let
Fz(q) := 1 fRf(~,w.x+q)dw. S1
Radon found that 0o
0
We refer to [GM] for the original 1917 paper and commentaries. An approximate implementation of (17) can be given by using the Fourier inversion formula 71-
y(x) ~ f Qo(x. ~)dO 0
where, as above, w = (cos 0, sin 0) and b
o(t) = f ITle2~t~:~l(Rf)(T)dT-b oo
"~ f I~-Ie2~'t~2:~(Rf)(T)dT" --00
(20)
1] This last approximation constitutes a band limiting process, and it can also be obtained from (14) as follows: Let wb be a "band-limiter', i.e, supp(~b) C [-b, b] and Wb = R#wb. Then (by letting g = wb in (14)), we obtain
Wb * f = R#(wb | R f) that is, we want Wb to be an approximate di-function choose Wb radial, e.g.,
(cf. [Na, Ch. 5]).
To begin with,
27r " b " where 0 < •(a) < 1, + = 0 for a > 1; this implies that wb = const. previous example is given by the ideal lowpass filter defined by
~'=
I~l+(~b ),
The
1 if0_l
and so
Wb(x) lbzJ~(blxl) =2~
(bill) '
where J1 is the Bessel function of the first kind and order one. We shall see below that one of the wavelet-based inversion formulas is inspired by (20). The formulas (15), (16), (17) allow for rather precise estimates of the degree in which the Radon transform and its inverse preserve the smoothness of the function f and data Rf. One way to measure this is to do it using Sobolev norms defined in an obvious way in the space of functions in the space of all lines. For instance, if f E C ~ ( B ) , where B is the unit disk in R 2, then one can find in [Na, Theorem 3.1], that for any real a, there are constants c, C > 0 such that cllf[lH~'(m) ~ Ilnf[[~.§
~ CIIflIH~'(B)
(21)
In the particular case of a = 0, we see that for f E L0~(R2) and supp(f) CC B one cannot expect better than control of one-half derivatives of Rf. Useful variations of the estimates (21) for other kinds of Radon transform can be found using that R#R is a Fourier integral operator of elliptic type [GS].
12 3. L o c a l i z a t i o n o f t h e R a d o n t r a n s f o r m
Returning to the problem of plasma physics that started these lectures, besides the fact t h a t the functions we are trying to detect seem to have a very large variation, that is, a large H 1 norm, we have the added difficulty that the amount of d a t a one can process or send down to Earth is fairly limited. One knows experimentally that, on a first approximation, all the variations from being Gaussian occur in the region where the values of f have gone down by 4-5 orders of magnitude. T h a t is, if f has a value 1 for the bulk velocity, then we are interested in the region where the values of f lie between 10 -4 to 10 -5 . The way this problem was traditionally solved (for conventional measuring devices) was to assume f had the form of a linear combination of a small number of Gaussians, and one just tries to estimate the variances and coefficients of these p e r t u r b a t i o n terms. If one does not want to impose these a priori restrictions on f , and we have only limited amount of d a t a to use, a natural idea is to just use those lines t h a t cross only the annular region where the main Gaussian varies between 10 -4 and 10 -s. (In the case of dimension 3 we would be dealing with a shell instead of an annulus.) This requires a localization of the Radon Transform. There are two ways to proceed. One, the most obvious (or naive) way is to try to localize the Radon transform as follows:
Reconstruct a function f in a disk B(a, r) from the data Rf(g), using only lines g passing through B(a, r). This cannot be true in dimension 2, as observed already by F. John. The reason is the well-known fact t h a t waves cannot be localized in 2-dimensions, namely, if we drop a pebble in the water, the ripples propagate along ever-expanding disks with time. In other words, an arbitrary perturbation confined to a disk at time t = 0 does not necessarily remain confined to the same disk (or any concentric disk, for t h a t m a t t e r ) at all future times. On the other hand, as F. John pointed out [J], if u is a solution of the wave equation A u - utt = 0, then its Radon transform v = Ru is a solution of the one dimensional wave equation vss - vu = 0, as it is seen i m m e d i a t e l y from the relation (8). For the one-dimensional wave equation with initial conditions at time t = 0, v(s, O) = Vo(S) and ~ 0) = vl(s), we have
v(8, t) =
1
I r~+t
(vo(s - t) + vo(s + t)) +
I S--t
If we think the pebble as being given by u(x,0) = no(s) = 0 for {xI _> c, 0 < ~ < < 1, and ut(x,O) = ul(x) =---O, then for a fixed w E S 1 and any later time t > 0 we would have with v(s,t) = R(u(.,t))(w, s) that vo(s) = v(s,O) = 0 for {sI > e while vl(s) = vt(s, 0) - 0. Thus, at any later time t > c we have that v(s, t) is only different from 0 for t - e < Is{ < t + e. Thus, the strict localization of the Radon transform would impose t h a t the support of u(x, t) be in the annulus t - ~ < {x I < t + e, which contradicts our observations. (Nevertheless, we shall see that some sort of localization takes place.) The other alternative, which fits the plasma problem, is to try to see whether we could reconstruct the values of f outside of a disk from the values of Rf(g), with never crossing t h a t disk. This turns out to be possible! It is the exterior problem for the Radon transform. We follow here the work of Quinto [Q]], [Q2] (and references therein.)
13 The starting point, as recognized in the pioneering work of A. M. Cormack, is to expand both the function f and its Radon transform g = Rf in a Fourier series. (For R n, n > 2, one uses spherical harmonics [Na, p. 25 ft.]) That is,
f(~) =
~
f~(r)~~~ x= (rcosO,rsinO),
l~--oo
Then, the Fourier coefficients ft and g~ are related by the two formulas oo
ge(s) =
2
2/Tl~l(~)(l-~)-~/2.fdr)dr,
(22)
8 O0
:~(r) --
~
~i
~ - ~)-~%i(~)d,,
(23)
T
where T~ is the Chebyshev polynomial of the first kind. One of the consequences of the Fourier Slice Theorem is that g cannot be an arbitrary function in the space of lines, it must satisfy certain compatibility conditions, usually called the moment conditions,
j s'~-lg(w, s)ds e
span{e 'k~ Ikl < m}, w = e ~.
This allows for a modification of (23) that makes it far more practical for numerical purposes [Na, p. 29-30]. This pair of equations show that the values of Rf(i) over all lines exterior to the disk B(0, r) are thought to determine f in the exterior of B(0, r). In particular, if one has supp f C B(0, P0), then the values of f in the annulus Pl < Ixl < P0 are entirely determined by the measurements of the Rf(~), only for lines ~ that intersect this annulus. (The uniqueness of the exterior problem and its variants is usually called the support theorem. It was first proved by Helgason in 1965, we refer to [Hell, [He2] for details and generalizations.) Quinto [Q1], [Q2] has successfully used this kind of ideas to obtain a very effective tomographic algorithm to determine cracks in the exterior shell of (usually large) circular objects, for instance, rocket nozzles. The method of Quinto is based on two things. First, the known characterization of the kernel of the exterior Radon transform in L 2 spaces with convenient radial weights (this is due to Perry for n -- 2 and Quinto for n > 3). For the case of interest at hand, n = 2, we consider the kernel of the exterior Radon transform in L2(B~, rdx), then the Fourier coefficients f~(r) must be given by the rule
fe(r) = For instance,
linear combination of
r 2-k, 0
~ k ~_ I~1, [~[- k even.
(24)
14 fo = O, f l = O , / 2 = c r - 2 , f3 = c r - 3 , f4 = c l r - 2 + c 2 r - 4 , . . .
The second observation is the fact that the Radon transform maps/-/1 := L2(B[, r(1 r)U2dx) into H2 = L2(S ~ x (1, cr g - ~ ) and one has an explicit diagonalization procedure for R, so that there are orthonormal bases ~j and Cj, respectively of//1 @ ker R and of ImR c H2, so that
R~j = and aj explicitly computable. support), we have
with aj > 0
gj~bj
Thus, for a given f of L2c(B~,dx) (i.e., of compact
oo
f = ~, aj~j + ]
with ] E ker R N n2c(S~,dx)
j--1
so that oo
R/=
~ ai~jCj j=l
thus 1 aj = - - < R f, Cj >H2 ~j This determines exactly ~ = f - ]. One expands now f in a Fourier series ~ f~(r)e ~e, with f~ of the form (22) as mentioned earlier. Now for r > > 1 we know that f -- 0, so that f = - ~ , thus, for r > > 1 we have f~(r) = - ~ e-~~176
but the
--Tr
coefficients f~(r) are polynomials in 1/r, so they are completely determined everywhere (up to r = 1) by their values for r > > 1. It is here that one uses a sort of analytic continuation, so it is fairly unstable, but Quinto has modified further this algorithm if one assumes f to be known in the small annulus 1 < Ix] < 1 + e, to give it further stability [Q2]. In the context of the plasma problem, we compared numerically the use of the same number of data measurements Rf(e), either spread throughout the whole disk versus the measurements taken only (and thus more densely) in the annulus of interest. We found the surprising result (to us) that the standard algorithm, with more thinly spread measurements did better. It was this numerical observation that led to the search of a different way to localize the Radon transform using wavelets. Let us first review briefly two other localization methods that had appeared earlier in the literature. The first one is the following. Let us assume that the unknown function f has support in the disk B1 of center 0 and radius 1, but that we are only interested on the values of f in Bb, 0 < b < 1, while we collect data on Ba, 0 < b < a < 1. (Note that all the disks are centered at 0). This is the situation considered for the interior Radon transform [Na, VI.4]. The basic idea is to make Rf(g) = 0 when e doesn't intersect B , and apply the standard reconstruction algorithm. In other words, we
]5 want to obtain (even approximately) the values f(x), IxlS b, from R(fxa), where Xa stands for the characteristic function of the disk B~. The first problem is that there are many non-zero functions f that have R(fx~) -- 0. Luckily, these functions do not vary much on Bb [Na], so one could just try to find f up to an additive constant (and try to find that constant by other means). One can see from the table or the formula (4.4) in [Na, p. 170], that one needs a = 4b to obtain a maximum L ~ error of 1.6% of the L 2 norm of f in B1. In particular, this procedure could not be applied if we are interested in f(x), for x E B(xo,a) C B1 with x0 close to cOB1. A typical such example is that of spinal chord studies. Usually, one study involves 40-60 CAT scans, that is, 40-60 scans along body sections perpendicular to the spine at different heights. The spinal chord area is about 15% of any such cross section of the body, and there would be a substantial reduction of radiation received by the patient if one localizes the CAT scan to only those lines passing through or near the spinal chord area. Another alternative that has been proposed is that of A-tomography [FRK], where one only attempts to reconstruct to discontinuities of the function f, i.e., perform edge detection in the image. The principle is based on the formula (15) namely, consider the "approximate" inversion
] = AR#Rf, so that / = 4~rAf. This formula preserves the "edges" (= discontinuities of f ) but not the actual values of f. A variation of this formula has been implemented in the Mayo Clinic to study angiograms [FRK]. Another interesting consequence of this kind of approximate formula is that it can also be applied to the attenuated Radon transform, oo
R,f(w, s) = / f(sw + tw•
§ tw •
?1))dr
-oo
where #(x,w) is assumed to be real analytic in R 2 • S 1 and nonnegative. This appears in SPECT tomography and, usually, both f and # are unknonwn. As observed by Kuchment and collaborators [KLM] the function A R # R j will have the same singularities as f. The point is that R#R, is still an elliptic Fourier integral operator. This fact had already been used effectively by many people, most notably Boman and Quinto [BQ], and it is the key observation in the work of Quinto [Q3], Ramm and Zaslavsky [RZ], and others. The method of localization we want to discuss here with a bit more detail is that of using wavelets to invert and localize the Radon transform in dimension 2. This general principle, which is joint work with David Walnut, was presented first in a 1990 NATO conference [W], and independently in [Ho]. Since then, similar ideas have appeared elsewhere in the literature (see, for instance, the recent volume [AU], the papers [BWl], [BW2], [DB], [DO], [O], and references therein.) True localization using discrete wavelets and filter banks is clearly developed in [FLBW]. (See also [FLB] for the fan beam case.) There are many excellent books on the subject of wavelets, at all levels of sophistication and different points of view, the following is a very partial list [M], [D],
16 [Ka]. There are actually two different, albeit related concepts, the continuous wavelet transform (CWT) (easier to understand) and the discrete wavelet transform (DWT) (easier to work with). The idea of CWT originates from the standard properties of the Fourier transform representation of nice functions. For f 6 L2(R) or f E S(R), we have both
f(~)
=
f f ( x ) e-2"ix~dx --00 O0
--00
and
II/II= IIfII: (
If(x)12dx) ~/2
-oo
If we translate f by b 6 R, Tbf(x) := f ( x -- b), then (vbf)'(~) = e2"ib~/(~), and for dilations we have D~f(x) := -~af(x/a)(a > 0), so that IIfJI2 = IID~f[12 and
(D~f)'(~) = D(1/~)/(~). In other words, the group x --+ ax + b (a > O, b q R) operates via unitary operators in L2(R), and has a corresponding representation on the space of Fourier transforms (which happens to coincide with L2(R)). The "problem" of the Fourier transform representation is that the behavior of f at a point ~ depends on the values of f everywhere, for that reason, the idea of a "windowed" Fourier transform has been introduced long ago, namely, introduce a cut-off function g (say, a "smooth" approximation of X[-1,1]) and consider
9T'l((Tbg)f)(~) = f g(x -- b)f(x)c-2"i~dx. -oo
Note that 5rl((7-bg)f)(~) is r 9 f(~), where r is the wavelet r = g(x)e2~iz~,(b(x) = r If we want to consider also the behavior at f at different scales we are led naturally to the CWT: Given a wavelet r 6 L2(R), and f 6 L2(R) we define 4-
A
../+
-oo
for 0 < a < cx), b 6 R, r denotes the complex conjugate of r and <, > denotes the L2-scalar product. We assume the wavelet is "oscillatory", that is, it is an arbitrary function in L2(R) which satisfies the condition
j
I (012 _
cr :=-oo - - - ~ a r
< oo.
17 OO
This condition implies that f r
= O. (For instance, when r is continuous at
--CO
= 0, which occurs if r E LI(R) N L2(R).) In fact, later on we will be interested in wavelets with many vanishing moments
f
xkr
= O, 0 < k < N .
--00
A typical wavelet is the Haar wavelet r = X[0,t/2] - X[t/2,t] so that
D1/2r = v~(X[0:/4]
-
X[1/4,1/2])
which shows that for k -+ co, D2-~r "analyzes" smaller and smaller details of the "signal" f. Moreover, Wcf determines f as seen from the following relation valid for any pair f,g C L2(R)
jf
dadb = cr < f,g > [1r
Wr
--00 --00
usually called Calderon's identity. If 11r property 1 Ilf-~
= 1 one also has the L2-approximation
Wcf(a, b)DaTbr
f
II ---~0
(26)
Al~lal<_A 2
Ibl_
as A2 ---+0+,A2 ~ +co, B --+ +co.
The generalization to R ~ is easy. A function r E L2(R ~) is a wavelet if
For a radial wavelet r E L2(R ~) and f C L2(R ~) we define the CWT by
Wr
f o r a e R \ ( 0 } , b C R ~,
where this time, D~r = lal-~/2r The interest of the CWT for tomograpy lies in the following two propositions from [BW2]. P r o p o s i t i o n 1. Let p r L2(R) be real valued, even, and satisfying OO
^
2
f IP(r)12dr < co
J 0
r3
(27)
18
Define a radial function r in R 2 by 3v2r
=
2~([~1)/1~1,then
r is a wavelet a n d
w,d(a, b) = a-'/2 f (w,P~/)(a, b. o) ) do3
(28)
Sa
P r o o f . Using the Fourier Slice Theorem we have for "y 9 R
It is then easy to verify that (27) implies that r is a wavelet in R 2. Recall that the Riesz transform of order a, I"~, of a function ~ 9 S ( R ) is defined by ( I ~ ) ' ( 9 ' ) -- h ' l - ~ ( 9 ' ) , thus the identity (29) can be rewritten as
p(t) = l:-~(P~r Extend p to a function in the space of lines by making it independent of the slope of the line, p(w, t) = p(t) for every w 9 S ~, then we have
= r
R#p(x) = 89162
since the last formula is a rewritting of formula (16) in terms of the Riesz transform. More generally, for any a > 0 and every w E S 1, we have
(R#Dapo,)(x) = aV2Df,(b(x), so that, using identity (14) and the fact that p is real valued, we obtain
Wcf(a,x)
= = =
(f.D.(p)(x) a-I/2(f. R#D~,:.,)(x) a-~/~R#(P~f | D~
= a-1/2/(WpP~f)(a,x.w)dw S1
This concludes the proof of the proposition.
9
A similar relation between the Radon transform and the C W T can be found using "separable" wavelets in R 2. P r o p o s i t i o n 2. Given a separable 2-dimensional wavelet of the form
r
= r162
where each r satisfies Ir < (71(1 + 171)-1 for all 7 9 R, define the family of one-dimensional functions {P~}~es' by 1
^1
^2
where w = (wl,w2) 9 S t. Then, for every f 9 L I ( R 2) VIL2(R2),
19
Gaussian
a n d Its H i l b e r t
transform
0.,
0.6
0.4
0.2
C
'\] i,j/
-0.2
-0.4' -10
I
~8
I
--6
I
--4
I
--2
I
I
J
I
I
0
2
4
6
8
10
Figure 3: Gaussian and its Hilbert transform.
(Wr
= a -1/2
J
f)(a,x . ~ ) d ~
s1
The point of Proposition 2 is the observation that the wavelet transform of a function f ( x ) with any mother wavelet and at any scale and location can be obtained by backprojecting the wavelet transform of the Radon transform of f using wavelets that vary with each angle, but which are admissible for each angle. So far we have not yet shown that the inversion formulas of the Radon transform based on wavelets do a good localization job. Using Proposition 1 the problem is clear, find a function p such that p has small support and simultaneously r has small support. From the relation (29) we see that we have overcome the Reisz operator of order - 1 , its symbol is 171 = (sgnv)v, so it is the composition of the differentiation and the Hilbert transform. (This is exactly the content of the inversion formula (18).) The problem, of course, is the Hilbert transform, but if we choose p with many vanishing moments, then we can overcome the difficulty. For the sake of comparison we show in Figure 3 the Hilbert transform of a Gaussian, its effective support is about four times the effective support of the Gaussian (defined by making zero those points below 1% of maximum value), which tails exactly with the result about the interior Radon transform mentioned earlier in this section. The key to explain the success of the wavelet method of localization is the following proposition [BW1], which in spirit is similar to the general principles about CalderonZygmund operators stated in [BCR]. P r o p o s i t i o n 3. Suppose that n is an even integer and the c o m p a ~ i y supported function h E L2(R) is such that for some integer m >_ 0 we have that h is n + ra - 1 times differentiable and satisfies
20 Relative exposure in a 256x256 pixels image 160
9O 80 ~, 70
19O Q
m 40
9Ol 20 10 i
L
20
40 60 80 10~ Radius of the region of interest in pixels
120
140
Figure 4: Exposure versus the radius of the ROI. (a) (b)
7jh(k)(7 ) E L~(R) N L2(R) ~ tJh(t)dt = 0
for 0 __ j _~ m, 0 < k < m + n - 1
for 0 < j _< m
--oo
Then = o(Itr ....
as ftl
and
t~+m-1II-~h C L2(R). The proof is rather elementary, it depends on the fact that if h is a function of compact support with m + 1 vanishing moments then 171~-1h(7) has n + m - 1 continuous derivatives. For ease of application it is better to work with the discrete wavelet transform (DWT). This is basically obtained by diseretizing the CWT or appealing to the multiresolution analysis of Mallat and Meyer [D], [M]. We have done this in detail in [FLBW] using coiflets [D] in order to be able to implement the inversion process using filter banks. One can show that to obtain a relative error of 0.5~o one only needs a margin of security of 12 pixels around the region of interest (ROI). For instance, to recover within this error bound an image occupying a disk of radius 20 pixels in a 256 • 256 image, one only needs about 25% of exposure, as shown in Figure 4. Figure 5 below is the Shepp-Logan phantom and its reconstruction from global fan beam data using the standard algorithm, in Figure 6 we use local data and our wavelets algorithm. The following figures are the reconstruction of a heart from real CAT scanner data using our wavelet method, and the reconstruction of the central part from local data and our wavelet method is found below.
21
(a)
(b)
Figure 5: (a) The Shepp-Logan head phantom; (b) the standard filtered backprojection in fan beam geometry (4).
Figure 6: Reconstruction from wavelet coefficients.
22
Figure 7: Reconstruction of heart from wavelet coefficients.
Figure 8: The local reconstruction of of central portion of heart.
23 We leave to the discussion and references in [FLBW] and [BW2] the comparison with other methods of inversion of the Radon transform using wavelets. One should add to the references in those two papers, the very recent work of Rubin [R], which is based on a systematic use of the Calderon reproducing formula and it is thus a development of the original ideas in [Ho].
24 4. T h e h y p e r b o l i c R a d o n t r a n s f o r m and E l e c t r i c a l I m p e d a n c e Tomography In this section we discuss the role tomography plays in a classical problem of Applied Mathematics, the inverse conductivity problem. Several of the earlier attempts to solve this problem involve generalizing the Radon transform to other geometries, that is, integrating functions over other families of curves beyond straight lines in the Euclidean plane. There are many examples of such transforms, in fact, the integration over great circles in S 2 was a transform considered by Minkowski and which inspired Radon in his work. The two we shall introduce presently are the generalized Radon transform of Beylkin [By] and the Radon transform on the hyperbolic plane [He1]. Let ~ be an open subset of R 2 and r E C ~ ( ~ • (R 2 \ (0))) be such that (a)
r
(b)
V=r
A~) ----Ar
~) for )~ > 0
r 0 for all (x,~) E ~ x (R 2 \ {0})
Then, for any s E R and w E S ~ we can define the smooth curve = {x e a : r
= s},
that is, the level curves of ~b. We let da denote the Euclidean arc length in such a curve. For u E C ~ ( ~ ) define the "Radon transform" R+u(~,s) = f
u(x)l V : r
5)ld,~(x)
/'/s ,~
Let h(x,~) be the Hessian determinant of r with respect to the second variables, 02 x h(x, ~) = d e t f ~ l then the "backprojection" operator Rr# is defined by h(x,~)
R v(x) = f
I
wES 1
Introducing K as the operator of convolution by 1/Ixl, Beylkin proved the following approximate inversion formula for the Radon transform as an operator R+: Lc2(a) --+ L~or namely,
R#cKRr = I + T
(30)
where, I is the identity map and T : L~(~) -+ L~or ) is a compact operator. In fact, Beylkin gives a recipe for a family of backprojection operators and generalized convolution operators K so that a decomposition of the type (30) holds. This gives his transform great flexibility and applicability to many problems, especially inverse acoustic problems, of course, the reader can easily verify
25 that for convenient choices of r the transform Re yields the Euclidean Radon transform studied earlier and the hyperbolic one, which we now introduce. (The reader should consult [Hell, [He3] for more details on this subject.) Let D, the unit disk of the complex plane C, be endowed with the hyperbolic metric of arc-length element ds given by 4'dzl~ ds2 - (1 -Iz[2) 2'
(31)
where Idzl denotes the Euclidean arc-length element. This metric is clearly conformal to the Euclidean metric but has constant curvature - 1 . The geodesics of this metric are the diameters of D and the segments lying in D of the Euclidean circles intersecting the unit circle COD perpendicularly. One can introduce geodesic polar coordinates z ++ (w, r), where w = z / H , r = d(z, 0). Note that Iz[ = tanh(r/2). In these coordinates the metric (31) can be rewritten as ds 2 = dr 2 + sinh 2 r dw 2 where dw 2 indicates the usual metric on cOD. The hyperbolic distance between two points is given by d(z,w)
arcsinh
Iz- !
((1-I~I~),~(I-l l )V
The Laplace-Beltrami operator AH on D can be written in terms of the Euclidean Laplacian A as
a.-
0-1zP)2a -
4 02 O 02 Or 2 + c o t h r + s i n h - ~ r o w 2.
(32)
The classical Moebius group of complex analysis is the group of orientation preserving isometrics of the hyperbolic plane D. One can define the hyperbolic Radon transform RH by Rf('~) = R H f ( 7 )
=
~ f ( z ) d s ( z ) , 7 geodesic in D
(33)
which is well defined for, say, continuous functions of compact support, or functions decaying sufficiently fast. Observe that to be integrable on the hyperbolic ray [0, oc[ (which is just the straight line segment from 0 to 1 in the complex plane C), f has to decay a bit faster than e -r. We denote by F the space of all geodesics in D, then the dual transform R # (or backprojection operator) is given by R#r
-- fr~ r
(34)
where Fz is the collection of geodesics through the point z and dttz is the normalized measure of Fz. Since a geodesic through z is determined by its starting direction w C S 1, then Fz ~ S 1 and d#z is naturally associated to ~ d w when we use this particular parameterization of Fz.
26 In order to invert R H one can proceed in the spirit of Radon's inversion formula (19). This was done by Helgason [He2, p. 155]. Or one can try to find a filtered backprojection type formula like (16). For that purpose we need to define convolution operators with respect to a radial kernel k. For k E L~oc([0, cxD)) and f E Co(D) we define f
k * f(z)
=
k
*H f(z) := ]D f(w)k(d(z, w))dm(w)
(35)
where dm(w) stands for the hyperbolic area measure, which in polar coordinates is given by
d m = sinh r drdw. Corresponding to the Euclidean formula (15) we have
R#HRHf : k * f, where k(t) -
1
Trsinh t
(36)
One can prove [BC1] that if
S(t) = cotht - 1
(37)
then
•41r
*H R
.R.
=
I
(38)
which is the exact analogue of (16). It is convenient to recall here that in the hyperbolic disk D we have a Fourier
transform [He2]. It is easier to work it out for "radial" functions as we interpret our kernel k, then the Fourier transform is defined with the help of the Legendre functions P~(r) by means of the following formula oo
k(A) = 2~rfk(t)P~_V2(cosht)sinhtdt
(A E R)
0
For radial functions k, m, we have (k * m)'(1) : k(~)~(~) So that, if/~(A) # 0 for all I E R, in principle, that is, for a convenient class of functions f, the convolution operator f , > k *H f is invertible. We refer to [He2], [BC1], [BC2], [Ku] for corresponding inversion formulas in the higher dimensional hyperbolic spaces, and the characterization of the range of the Radon transform. In particular, [Ku], [BC2] exploit the "intertwining" between RH and the Euclidean Radon transform as well as the Minkowski-Radon transform on spheres. Let us cxplain now what the above hyperbolic Radon transform has to do with Electrical Impedance Tomography (EIT) and what EIT is. Let us consider the following tomographic problem: using a collection of electrodes of the type used in electrocardiograms (EKG) uniformly distributed around the breast
27 of a patient and all lying in the same plane, introduce successively (weak) currents at each one of the electrodes (as done in EKG) and measure the induced potential at the remaining ones. The objective is to obtain an image of a cross section of the lungs to determine whether there is a collapsed lung or not. This was what Barber and Brown set up to do in 1984 [BB1], [BB2]. The point being that this equipment is cheap, transportable and provides a non-intrusive test (that is, no punctures have to be done to the chest cavity). Similarly, one can try to determine the rate of pumping of the heart using this kind of equipment. Notc that the pulse only determines the rate of contracting and expanding of the heart but not how much blood is being pumped by it. Another completely different problem arises in the determination of the existence and lengths of internal cracks in a plate, by using electrostatic measurements on the boundary [FV], [BCW], [W]. These three are examples of the following inverse problem. (The best reference for the general facts about this problem is the supply [SU]. See also the nice explanation for the general public [C], [S]): Assume/5 is a strictly positive (nice) function in the closed unit disk D. If we were to introduce a current at the boundary OD, represented by a function ~ satisfying fOBCds = 0, then the Neumann problem div (/3 g r a d u ) /3~
=0inD = r on
OD
(39)
has a solution u which is unique up to an additive constant. If r is a nice function then ~ (that is, the tangential derivative of u) is well defined on OD, so we have the input-output map 0u which is a linear continuous map from the Sobolev space H~(OD) into itself. (This statement holds for any domain D with nice boundary, not just the disk.) Consider now the (very non-linear) map /3,
>A/~
(40)
is it injective? Can one find the inverse to this map? This problem was originally posed by A. Calder6n, who proved that (40) was locally invertible near/3 = constant, more recently Nachman IN1], IN2] proved global invertibility. Since/3 is usually called the conductivity and 1//3 the impedance, this is the reason for the name EIT of this inverse problem. In the biological applications we know the value/3 for the different constituents like blood, lung tissue, etc., so one only looks for a profile of the areas occupied by them. In the determination of cracks, one can assume/3 "known", except for curves where/3 = 0, and one wants to determine this curve, or whether any exists. One can find in [SU] many important inverse problems that are equivalent to EIT: in acoustics, radiation scattering, etc. Note that in the problem of the rate of pumping of the heart, we can think that all we want to determine is just a single number, this rate. Isaacson, Newell and collaborators have in fact patented [C], [I] a device that measures this rate with the help of EIT. We also know that this problem, being an inverse elliptic problem is very ill-conditioned, so in any case one is willing to restrict oneself to find the deviation of/3 from an assumedly known conductivity/30. In the simplest case we assume/30 --= 1, so that/3 = 1 +5/3, 15/31 < < 1, and we further assume
28 5/3 = 0 on OD (One can always reduce matters to this case). Thus u = U + 5U, where U is the solution of (39) for the same boundary value, and/3 = 1. In other words AU OU
= 0 = r
in D on OD
(41)
Here A is the Euclidean Laplacian. The perturbation 5U then satisifes A(SU) o(~u) o~
= - < grad (&3), grad U > in D -- -(8/~)~b on OV
We have at our disposal the choice of inputs r
(42)
Their only restriction is that
rOD ~bd8 : O. For that reason, they can be well approximated by linear combinations
of dipoles. A dipole at a point oo c OD is given by It turns out that the solution Uw of { AU~ = 0 -r~~
o
inD on cOD
(43)
has level curves which are arcs of circles passing through w and perpendicular to 0D. That is, the level curves of U~, are exactly the geodesics of the hyperbolic metric. This fact passd unnoticed to Barber and Brown but they definitely realized that the value O(SU) (44) #Os at a point a E OD must be some sort of integral of 53 over the level curve of U~ that ends at a, precisely the geodesic starting at w and ending at a. In other words, # is a function in the space of geodesics in D considered as the hyperbolic plane, all the geodesics are obtained this way by changing w and a. Without expressly stating this, Barber and Brown introduced a "backprojection" operator that turned out to be exactly R#H and gave the approximation to ~ as R#H#. Santosa and Vogelius recognized explicitly that some sort of Radon transform was involved and used the generalized Byelkin transform and a convenient choice of K in (30) to stabilize numerically the inversion of EIT. Casadio and I, prodded by a question of Santosa and Vogelius, saw that RH was involved and developed the inversion formula (38) for this purpose. As it turns out, all of these approaches are just approximations to the linearized problem. Only in [BC3], [BC4], we realized the fact that the exact formulation of the linearized problem in terms of hyperbolic geometry requires also a convolution operator! Namely, let ~r
= c~
- 3 cosh-4(t) (45) 8r and # the boundary data (44) considered as a function on the space of geodesics in D, then one has that the exact relation between 5/? and # is given by
RH(a *H ~/~) = #
(46)
Using the backprojection operator we also obtain R# # = R# RH(g *H ~1~)
(47)
29 so that 1 G a H ( S *H (n~,,)) = ~ *H 6fl
(48)
which requires to invert the convolution operator of symbol ~. One can compute its hyperbolic Fourier transform k exactly and find out that k(.~) :fi 0 for every A E R, so that the operator ~. is, in principle, invertible, but the numerical implementation of this inversion has proven difficult so far. (Although Kuchment and his students have made in [FMLKMLPP] some progress towards implementing a numerical Fourier transform in D, which we hope will prove useful to compute 6ft.) One can recognize in (47) and (48) the same principle that lead to the numerical approach in [BB1], [SV] and others. Due to the importance of this problem there have been many other interesting approximate inversion formulas, under special assumptions on the conductivity fl, for instance, fl is "blocky", that is the linear combination with positive coefficients of a finite number of disjoint squares [DS]. Their approach is variational, and one may wonder whether one could not use some version of the Mumford-Shah edge detection algorithms [MS] to obtain a rather sharp solution of the inverse conductivity problem (40). 5. F i n a l r e m a r k The objective of these short notes (and the corresponding CIME course) was only to indicate how, beyond the well-known applications of tomography to Medicine, there are many other possible ones. Moreover, even to solve them approximately, they require deep mathematical tools, showing once more that the applicability of "pure" and "abstract" mathematics is not a fairy-tale but a concrete reality. It also indicates that it pays to "invest" one's time trying to communicate with those, be they physicists, or physicians, etc., that have the ready made applications. A lesson often lost by graduate students in Mathematics.
30 6. R e f e r e n c e s [AA] S. Andrieux and A. Ben Alda, Identification de fissures planes par une donn~e de bord unique, C.R. Acad. Sci. Paris 315 I(1992), 1323-1328. [AU] A. Aldroubi and M. Unser, editors, "Wavelets in Medicine and Biology," CRC Press, 1966, 616 pages. [BB1] D. C. Barber and B. H. Brown, Recent developments in applied potential, in "Information processing in Medical Imaging," S. Bacharach (ed.), Martinus Nijhoff, 1986, 106-121. [BB2] D. C. Barber and B. H. Brown, Progress in Electrical Impendance Tomography, in "Inverse problems in partial differential equations," D. Colton et al. (eds.), SIAM, 1990, 151-164. [BC1] C. A. Berenstein and E. Casadio Tarabusi, Inversion formulas for the kdimensional Radon transform in real hyperbolic spaces, Duke Math. J. 62 (1991), 613-632. [BC2] C. A. Berenstein and E. Casadio Tarabusi, Range of the k-dimensional Radon transform in real hyperbolic spaces, Forum Math. 5 (1993), 603-616. [BC3] C. A. Berenstein and E. Casadio Tarabusi, The inverse conductivity problem and the hyperbolic x-ray transform, in "75 years of Radon transform," S. Gindikin and P. Michor, editors, International Press, 1994, 39-44. [BC4] C. A. Berenstein and E. Casadio Tarabusi, Integral geometry in hyperbolic spaces and electrical impedance tomography, SIAM J. Appl. Math. 56 (1996), 755-764. [BCW] C. A. Berenstein, D. C. Chang and E. Wang, A nondestructive inspection method to detect a through crack by electrostatic boundary measurements, ISRTR 96-1. [BW1] C. A. Berenstein and D. Walnut, Local inversion of the Radon transform in even dimensions using wavelets, in "75 years of Radon transform," S. Gindikin and P. Michor, editors, International Press, 1994, 45-69. [BW2] C. A. Berenstein and D. Walnut, Wavelets and local tomography, in "Wavelets in Medicine and Biology," A. Aldroubi and M. Unser, editors, CRC Press, 1966. [BQ] J. Boman and E. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J. 55 (1987), 943-948. [BV] K. M. Bryant and M. Vogelius, A computational algorithm to detect crack locations from electrostatic boundary measurements, Int. J. Eng. Sci. 32 (1994), 579-603. [By] G. Beylkin, The inversion problem and applications of the generalized Radon transform, Comm. Pure Appl. Math. 37 (1984), 579-599.
31 [BCR] G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991), 141-183. [C] B. Cipra, Shocking images from RPI, SIAM News, July 1994, 14-15. [D] I. Daubechies, "Ten lectures on wavelets," SIAM, 1992. [DB] A.H. Delaney and Y. Bresler, Multiresolution tomographic reconstruction using wavelets, ICIP-94, 830-834. [DO] J. DeStefano and T. Olson, Wavelet localization of the Radon transform, IEEE Trans. Signal Proc. 42 (1994), 2055-2057. [DS] D. C. Dobson and F. Santosa, An image enhancement technique for electrical impedance tomography, Inverse Problems 10 (1994), 317-334. [FRK] A. Faridani, E. Ritman and K. T. Smith, Local tomography, SIAM J. Applied Math. 52 (1992), 1193-1198. [FLBW] F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein and D. Walnut, Waveletbased multiresolution local tomography, ISR-TR 95-73, see also ICIP-95, Washington, DC. [FLB] F. Rashid-Farrokhi, K. J. R. Liu and C. A. Berenstein, Local tomography in fan-beam geometry using wavelets, ICIP-96, Laussane. [FMP] B. Fridman, D. Ma, and V. G. Papanicolau, Solution of the linearized inverse conductivity problem in the half space, preprint Wichita St. U., 1995. [FMLKMLPP] B. Fridman, D. Ma, S. Lissianoi, P. Kuchment, M. Mogitevsky, K. Lancaster, V. Papanicolaou, and I. Ponomaryov, Numeric implementation of harmonic analysis on the hyperbolic disk, in preparation. [FV] A. Friedman and M. Vogelius, Determining cracks by boundary measurements, Indiana U. Math. J. 38 (1989), 527- 556. [GG] I. M. Gelfand and S. Gindikin, editors, "Mathematical problems of tomography," AMS, 1990. [GM] S. Gindikin and P. Michor, editors, "75 years of Radon transform," International Press, 1994. [GIN] D. Gisser, D. Isaacson, and J. Newell, Current topics in impedance imaging, Clin. Phys. Physiol. 8 (1987), 216-241. [GS] V. Guillemin and S. Sternberg, "Geometric asymptotics," AMS, 1977. [He1] S. Helgason, "The Radon transform," Birkh~user, 1980. [He2] S. Helgason, "Groups and geometric analysis," Academic Press, 1984. [He3] S. Helgason, "Geometric analysis on symmetric spaces," AMS, 1994.
32 [Ho] M. Holschneider, Inverse Radon transform through inverse wavelet transforms, Inverse Problems 7 (1991), 853-861. [J] F. John, "Plane waves and spherical means," Springer-Verlag, reprinted from originial edition Interscience, 1955. [Ka] G. Kaiser, A friendly guide to wavelets, Birkhguser, 1994. [KS] A. C. Kak and M. Slaney, "Principles of computerized tomographic imaging," IEEE Press, 1988. [KaS] P. G. Karp and F. Santosa, Non-destructive evaluation of corrosion damage using electrostatic measurements, preprint 1995. [KR] A. I. Katsevich and A. G. Ramm, New methods for finding values of jump of a function from its local tomography data, Inverse Probl. 11 (1995), 1005-1023. [Ke] F. Keinert, Inversion of k-plane transforms and applications in computer tomography, SIAM Riview 31 (1989), 273-289. [KLM] P. Kuchment, K. Lancaster and L. Mogilevskaya, On local tomography, Inverse Problems 11 (1995), 571-589. [KSh] P. Kuchment and I. Shneiberg, Some inversion formulas for SPECT, Applicable Analysis 53 (1994), 221-231. [Kul] A. Kurusa, The Radon transform on hyperbolic space, Geometriae Dedicata 40 (1991), 325-336. [Ku2] A. Kurusa, Support theorems for the totally geodesic Radon transform on constant curvature spaces, Proc. Amer. Math. Soc. 122 (1994), 429-435. [M] Y. Meyer, "Ondelettes et op&ateurs," 3 vols., Herman, 1990. [MS] J. M. Morel and S. Solimini, "Variational methods in image segmentation," Birkhguser, 1995. [N1] A. I. Nachman, Reconstruction from boundary measurements, Annals Math. 128 (1988), 531-576. IN2] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Annals Math. 143 (1996), 71-96. [Na] F. Natterer, "The mathematics of computerized tomography," Wiley, 1986. [O] T. Olson, Optimal time-frequency projections for localized tomography, Annals of Biomedical Engineering 23 (1995), 622-636. [Q1] E. T. Quinto, Tomographic reconstruction from incomplete data-numerical inversion of the exterior Radon transform, Inverse Problems 4 (1988), 867-876. [Q2] E. T. Quinto, Singularities of the X-ray transform and limited data tomography in l:t2 and R 3, SIAM J. Math. Anal. 24 (1993), 1215-1225.
33 [Q3] E. T. Quinto, Computed tomography and rockets, Springer Lecture Notes in Math. 1497 (1991), 261-268. [QCK] E. T. Quinto, M. Cheney, and P. Kuchment, eds., "Tomography, impedance imaging, and integral geometry," Lect. Appl. Math. 30, Amer. Math. Soc., 1994. [R]B. Rubin, Inversion and characterization of Radon transforms via continuous wavelet transforms, Hebrew Univ. TR 13, 1995/96. [RS] A. Ramm and A. I. Zaslavsky, Singularities of the Rdaon transform, Bull. Amer. Math. Soc. 25 (1993), 109-115. IS] F. Santosa, Inverse problem holds key to safe, continuous imaging, SIAM News, July 1994, 1 and 16-18. [ST] H. Schonberg and J. Timmer, The gridding method for image reconstruction by Fourier transformation, IEEE Trans. Medical Imaging 14 (1995), 596-607. [SCII] E. Sommersalo, M. Cheney, D. Isaacson, and I. Isaacson, Layer stripping: a direct numerical method for impedance imaging, Inverse Probl. 7 (1991), 899-926. [SU] J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications, in "Inverse problems in partial differential equations," D. Colton et al., eds., SIAM, 1990, 101-139. [SV] F. Santosa and M. Vogelius, A backprojection algorithm for electrical impedance imaging, SIAM J. Appl. Math. 50 (1990), 216-243. [W] D. Walnut, Applications of Gabor and wavelet expansions to the Radon transform, in "Probabilistic and stochastic methods in analysis," J. Byrnes et al., ed., Kluwer, 1992, 187-205. [Wa] E. Wang, Ph.D. thesis, University of Maryland, College Park, 1996. [ZCMB] Y. Zhang, M. A. Coplan, J. H. Moore and C. A. Berenstein, Computerized tomographic imaging for space plasma physics, J. Appl. Phys. 68 (1990), 58835889.
Institute for Systems Research University of Maryland College Park, MD 20742 [email protected]
HOLOMORPHIC ANALYTIC
MAPPINGS
SUBMANIFOLDS
BETWEEN IN COMPLEX
REAL SPACE
PETER EBENFELT
Department of Mathematics Royal Institute of Technology I00 44 Stockholm Sweden 1996-9-28 CONTENTS 1. Introduction 2. Preliminaries on algebraic mappings and CR geometry 2.1. Algebraic mappings 2.2 Real analytic CR submanifolds in C N 2.3 Finite type and minimality 2.4 Normal forms for generic submanifolds 2.5 The complexification of a real analytic submanifold 3. Holomorphic nondegeneracy, finite nondegeneracy, and reflection identities for holomorphic mappings 3.1. Holomorphic nondegeneracy of real analytic CR submanifolds 3.2. Finite nondegeneracy of real analytic CR submanifolds 3.3. Reflection identities for holomorphic mappings 4. The Segre sets 4.1. The Segre sets of a real analytic CR submanifold 4.2. Homogeneous submanifolds of CR dimension 1 4.3. Proof of Theorem 4.1.21 (CR dimension 1 case) 5. An application to holomorphic mappings between real algebraic submanifolds 5.1. A reformulation of Theorem 1.1 5.2. Proof of Theorem 5.1.1 6. Other applications and concluding remarks 6.1. The algebraic equivalence problem 6.2. Uniqueness of biholomorphisms between real analytic, generic submanifolds
36 1. INTRODUCTION In this paper we shall present, in fairly self-contained form, some recent ideas and concepts concerning real analytic submanifolds in C N. One of the main constructions described here is an invariant sequence of sets--called the Segre sets--attached to a real analytic submanifold in cN; this sequence of sets was introduced in joint work by the author together with Baouendi and Rothschild [BER1]. The first Segre set coincides with the so-called Segre variety, introduced by Segre [Seg] and successfully used in mapping problems for real analytic hypersurfaces by a number of authors: e.g. Webster [W1], Diederich-Webster [DW], Diederich-Fornaess [DF], Huang [H2], and others. Subsequent Segre sets turn out to be unions of Segre varieties. One of the merits of the Segre sets is that they allow one to analyze "reflection identities" (see section 3.3) for higher codimensional submanifolds--the idea of using reflection identities to analyze CR mappings goes back to e.g. Lewy ILl, Baouendi-Jacobowitz-Treves [BJT], Baouendi-Rothschild [BR1]. The Segre sets also allow a new characterization of the notion of finite type, as introduced by Bloom-Graham [BG]. In fact, the Segre sets provide a way of constructing the CR orbits of a real analytic CR submanifold without solving any differential equations (see Theorem 4.1.21). As a result, one finds e.g. that the CR orbits of a real algebraic CR submanifold are algebraic (Corollary 4.1.31). As an application and illustration of these techniques, we shall prove the following result (which we shall also reformulate in terms of more c]assical CR geometry, Theorem 5.1.1) from [BER1]. T h e o r e m 1.1 ( [ B E R 1 ] ) . Let A C C N be an irreducible real algebraic set, and Po a point in A such that Po C Ares. Suppose the following two conditions hold. (1) There is no hoIomorphic vector field (i.e. a vector field with holomorphic coe~cients and values in T I ' ~ which is tangent to an open piece of Areg. (2) If f is a germ, at a point in A, of a holomorphic ]unction in C N such that the restriction of f to A is real valued, then f is constant. Then, if H is a holomorphic map from an open neighborhood in C N of po E A into C N, with Jac H ~ O, that maps A into another real algebraic set A' with dim A' = dim A, necessarily the map H is algebraic.
The first result along these lines goes back to Poincar6 [P] who proved that a biholomorphic map H : C 2 ~-* C 2 defined near a point on a sphere S C C 2 and mapping S into another sphere S' C (;2 is rational. This was later extended to mappings between spheres in C g by Tanaka ITs]. Webster [W1] then proved that a biholomorphic mapping H : C g H C N, defined in some open subset of C N, taking a real algebraic, somewhere Levi-nondegenerate (i.e. with nondegenerate Levi form at some point) hypersurface M into another real algebraic hypersurface M ' is algebraic. Recently, Baouendi Rothschild [BR3] showed that if the real algebraic hypersurface M satisfies condition (1) of Theorem 1.1 above (which is a weaker condition than being Levi nondegenerate somewhere) then any holomorphic mapping H : C N ~-* C N, defined in some open subset of C N and with Jac H ~ 0, taking M into another real algebraic hypersurface M I is algebraic. Moreover, they show
37 that this condition is also necessary for such a conclusion to hold in the sense that if (1) is violated then there is a non-algebraic biholomorphism of M into itself. The sufficiency of condition (1) in this result by Baouendi-Rothschild is contained in Theorem 1.1 above, because a real analytic hypersurface that satisfies condition (1) automatically satisfies condition (2). We would also like to mention that the conditions (1) and (2) in Theorem 1.1 are essentially necessary for the conclusion of the theorem to hold. We refer the reader to [BER1] for details on this (see also section 6). Condition (1) was first introduced, and named holomorphic nondegeneracy (see section 3.1 for a detailed treatment of this notion), by Stanton [Stl] in connection with the study of infinitesimal CR automorphisms of real hypersurfaces. It deserves to be mentioned here that holomorphic nondegeneracy is fairly easy to verify because it turns out to be closely related to another property (finite nondegeneracy, see section 3.2), which is very computational and is a direct generalization of Levi nondegeneracy. Results of the type above for mappings between hypersurfaces in different dimensional spaces have been obtained by e.g. Webster [W2], Forstneric IF], and Huang [H1]. For higher codimensional submanifolds, work has been done by e.g. Senkin-Tumanov [TH], Tumanov [Wu2], and Sharipov-Sukhov [SS]. Other applications to rigidity properties of holomorphic mappings between real analytic submanifolds will be briefly discussed in the last section of this paper. The paper is organized as follows. In chapter 2, we give the basic definitions and facts, mostly without proofs, concerning algebraic mappings and CR geometry. More recent concepts such as holomorphic nondegeneracy and finite nondegeneracy, as well as reflection identities for holomorphic mappings, are introduced and discussed in chapter 3. The construction of the Segre sets and proofs of the main results concerning these are given in chapter 4. In chapters 5-6, applications of the techniques are discussed. A reformulation of Theorem 1.1 and a proof is given in chapter 5. Applications to uniqueness questions and some open problems are discussed in the final chapter. A c k n o w l e d g e m e n t . The author would like to thank Professors M. S. Baouendi and L. P. Rothschild for agreeing to have results and arguments from our joint papers [BER1-3] included in these notes. As the reader will no doubt notice, the results presented here, for which the author can claim any credit, are due to this above mentioned joint work. 2. PRELIMINARIES ON ALGEBRAIC MAPPINGS AND C R GEOMETRY 2.1. A l g e b r a i c m a p p i n g s . We denote by ON(po) the ring of germs of holomorphic functions in C g at p0, and by .AN (P0) the subring of ON (Po) consisting of those germs that are also algebraic, i.e. those germs for which there is a nontrivial polynomial P(Z,x) e C[Z,x] (with Z E C N and x C C) such that any representative f(Z) of the germ satisfies (2.1.1)
P(Z, f(Z)) =_O.
In particular, any function in .AN(Po) (throughout this paper we shall, without comment, identify a germ of a function with some representative of it) extends
38 as a possibly multi-valued holomorphic function in C N \ V, where V is a proper algebraic variety in C N. We list here some basic properties of algebraic holomorphic functions that will be used in the proof of Theorem 1.1. We use the notation AN for AN(O). L e m m a 2.1.2.
The following holds:
(i) If f E `AN then O~"f E r for any multi-index a. (ii) If f G `AlV and gj E ,AN with gj(O) = O, for j = 1, ..., I(, then
f(gl(Z), ...,9K(Z)) E `AN. (iii) (The Algebraic Implicit Function Theorem) Let F( Z, x) be an algebraic holomorphic function near 0 in C y x C, i.e. F E .,4N+1, and assume that
F(o, o) = o
,
OF -g-;(o, o) # o.
Then there is a unique function f ~ .AN such that x = f ( Z ) solves the equation F ( Z , x ) = O, i.e.
F(Z, f ( Z ) ) = 0 The arguments needed to prove this lemma are standard (see e.g. [BM], and also [BR3] for further properties of algebraic functions), and the proof is omitted. We say that a germ of a holomorphic mapping H : C N --~ C K at p0 is algebraic if the components of H (we write H = (H1,..., Hh')) are all algebraic. It follows from Lemma 2.1.2 (ii) that this property is invariant under algebraic changes of coordinates in C x and C K at P0 and p~ = H(p0), respectively.
2.2. R e a l a n a l y t i c C R s u b m a n i f o l d s in C N. In sections 2.2-2.5, we shall set up the notation, and give the basic definitions and results from CR geometry needed for subsequent sections. Most facts and results in these sections will be stated informally, and without proofs. Unless otherwise specified, proofs can be found in e.g. [B]. We should point out that only real analytic submanifolds will be considered. The definitions presented in these sections can be made in the broader category of smooth (e.g. C ~176submanifolds, but some of the facts stated fail to be true in that general setting. For instance, the two notions "finite type" and "minimality" presented in section 2.3 coincide for real analytic CR submanifolds, but do not coincide in general for merely smooth CR submanifolds. Let M be a real analytic submanifold in C N and P0 a point in M. Let m be the (real) codimension of M. We may describe M near P0 as the zero locus
(2.2.1)
M = {Z c cN: p(z, 2) = 0},
where p = (Pl ..... Pro) are real valued, real analytic functions near P0 with linearly independent differentials dpl, ..., dpm; we use the notation h(Z, 2) for a real analytic function in C N to indicate that we think of such objects as restrictions to ~ = Z of
39 holomorphic functions of (Z, 4) 6 C N • C N. We say that M is real algebraic if the pj can be taken to be real valued polynomials in Z and 2. The complex tangent space of M at p 6 M is defined as (2.2.2)
T;(M) = Tp(M) A Jp(TR(M))
where Jp: Tp(C N) ~ Tp(C N) denotes the complex structure in C N. The (real) dimension of Tp(M) is even and satisfies (2.2.3)
2 N -- 2m <_ d i m ~ T ; ( M ) < 2N - m.
If M is a hypersurface, i.e. a real codimension 1 submanifold, then d i m i T y ( M ) is necessarily 2N - 2. In general, the dimension varies with p 6 M. D e f i n i t i o n 2.2.4. M is said to be CR at p G M if dim~Tq(M) is constant for all
q in some neighborhood ofp in M. We decompose d as d = 0 + c3. The real analytic submanifold M is CR at p 6 M if and only if the rank of the covectors Opl, ..., Opm is constant at all q 6 M near p. Hence, any real analytic submanifold M C C N is CR outside a proper real analytic subset V C M. We say that a connected submanifold is CR if it is CR at every point. The complexified complex tangent space CTp(M) = C | T;(M) decomposes as the direct sum T(I'~ (2.2.5)
+ T(~
where
T(I'~
=
CTp(M) A Tp(l'~
T(~
: CTp(M) N T(~
N)
here, T ( I ' ~ and T(~ N) denote the spaces of (1,0)-vectors and (0,1)vectors in C N respectively, and CTp(M) denotes the eomplexified tangent space of M , G | Tp(M). If M is CR then (2.2.6)
T(~
= O Tp( ~ pcM
forms a complex vector bundle over M, called the CTl bundle. Sections of the CR bundle are called CR vector fields. A function (or distribution) defined on (a piece of) M is said to be a CR function (or CR distribution) if it is annihilated by all the CR vector fields on (that piece of) M. The restriction to M of a holomorphic function defined in a neighborhood of M is CR, and any real analytic CR function on M is the restriction of such a holomorphic function. In general, there are smooth (C ~ ) CR functions on M that are not restrictions of holomorphic functions. D e f i n i t i o n 2.2.7. M, of codimension m, is said to be generic at p E M if the dimension of T;(M) is minimal, i.e. d i m ~ T ; ( M ) = 2 N - 2m. The real analytic submanifold M is generic at p 6 M if and only if the rank of the covectors Opl,...,Opm equals m at p. In particular, if M is generic, then it is also CR.
40 If M is CR, we call the complex dimension of T(~ ( = d i m i T y ( M ) / 2 ) the CR dimension of M and the real dimension of T~(M)/T;(M) the CR codiraension of M. For a generic submanifold M C C N, the CR codimension equals the codimension rn and (2.2.8)
N = CR dim (M) + CR codim (M).
Another important fact is that a real analytic CR submanifold M is generic if and only if it is not contained in a proper complex submanifold of C N. If _~4 is not generic, then there is proper complex submanifold 2( C C N (unique if we consider X as a germ of a submanifold) such that M is generic as a real analytic submanifold of X. We shall refer to X as the intrinsic complexifieation of M. The Levi form of a CR submanifold M at p ~ M is the (vector valued) quadratic form on T(p~ defined as follows (2.2.9)
T(p~
~ Lp ~ ~r,([L,L]) E CTp(M)/CT;(M),
where rr, is the projection 7r,: CTp(M) ~ CTp(M)/CT;(M) and L is some section of T(~ that equals Lp at p. A real hypersurface M is said to be Levi nondegenerate at p E M if the quadratic form (2.2.9) is nondegenerate. In this paper, we shall not impose any conditions on the Levi form of a C R submanifold. Instead, we shall introduce (in section 3.1) a weaker condition, which also turns out to be essentially necessary in most applications we shall consider. We conclude this section by giving a few examples of generic, CR, and non-CR submanifolds. E x a m p l e 2.2.10. A complex subrnanifold of C N is CR but not generic. E x a m p l e 2.2.11. A real hypersurface in C N is generic. E x a m p l e 2.2.12. Consider the real analytic 4-dimensional submanifold M C C 3 defined by (2.2.13)
ImZ3-lZl]
2-lZ212=0
,
ImZ2=0.
If we write pl(Z, Z) = 0 for the first equation and p2(Z, 2 ) = 0 for the second, then it is easy to verify that
(2.2.14)
Off1 = ~1d Z 3 - 21dZ1 - Z~dZ2 ,
Op2 = ldz2. 2
The rank is two and, hence, M is generic. E x a m p l e 2.2.15. Consider the real analytic 2-dimensional submanifold M C C 2 defined by (2.2.16)
Im Z~ -]Z112 = 0
,
Re Z2 = 0.
As above, we write p l ( Z , Z ) = 0 for the first equation and p2(Z,Z) = 0 for the second. We find that (2.2.17)
Opl = ~dZ2 - 2~dZ~
,
Op2 =
dZ2.
Outside {Z~ --- 0} the rank is two, and along {Z1 --= 0} the rank is one. Thus, M is generic outside the origin but is not even CR at the origin.
41 2.3. F i n i t e t y p e a n d m i n i m a l i t y . Let M C C N be a real analytic C R submanifold.
Definition 2.3.1. M is said to be of finite type at Po E M if the CR vector fields, the complex conjugates of the CR vector fields, and their commutators, evaluated at Po, span CTp0(M ). Equivalently, M is said to be of finite type at P0 E M if the sections of TO(M) and their commutators, evaluated at p0, span Tp0 (M). More generally, we define the Harmander numbers of M at P0 E M as follows. We let E0 = T~o(M) and Pl the smallest integer > 2 such that the sections of TO(M) and their commutators of lengths < #1 evaluated at P0 span a subspace E1 of Tpo(M ) strictly bigger than E0. The multiplicity of the first Hhrmander number #1 is then gl = dim~E1 - dim~E0. Similarly, we define/*2 as the smallest integer such that the sections of of TO(M) and their commutators of lengths
Definition 2.3.2. M is said to be minimal at Po E M if M contains no proper CR submanifold through po with the same CR dimension as M . We define the (local) CR orbit of po in M as the Nagano leaf of the CR vector fields through P0. The CR orbit of p0 is a minimal CR submanifold Wpo C M through P0 such that T;o(M ) C Tpo(Wpo). In fact, using the notation above, the tangent space Tpo(Wpo ) equals the space E,. For a real analytic CR submanifold M , the notions introduced above are related as follows: M is minimal at po ~=:=> Wp0 contains an open neighborhood of P0 in M r M is of finite type at P0. We refer the reader to [Tul] and the paper by ~hamartov in these Proceedings for further reading on minimality and its connection to wedge extendibility of CR functions. E x a m p l e 2.3.3. Consider the real analytic hypersurface M C C 2 defined by (2.3.4)
Im Z2 = (Re
Z2)IZll 2.
Note that the complex hyperplane {Z2 = 0} is contained in M. It follows that M is not minimal (not of finite type) at the points (Z1,0) E M. However, M is minimal (of finite type) at all other points.
42 If M is a real anMytic hypersurface and M is not minimal at a point P0 E M , then M contains a complex hypersurface through t h a t point. This follows from the fact t h a t the C R orbit of P0 in M , being a proper C R submanifold of M with C R dimension N - 1 , necessarily has dimension 2 N - 2 and, hence, is a complex manifold by the N e w l a n d e r - N i r e n b e r g theorem. Also, if a real analytic hypersurface M is not minimal at most points (outside a proper analytic subvariety), then M is Levi flat (i.e. M is locally biholomorphically equivalent to a real plane or, which is the same, the Levi form of M vanishes identically). Both of these facts fail in general for higher codimensionat C R submanifolds as the following example illustrates. E x a m p l e 2.3.5. Consider the codimension two, real analytic, generic submanifold M C C a defined by (2.3.6)
Im
Z 3 ---- I Z 1 1 2
Im Z2 = 0.
,
M is foliated by the C R submanifolds Nx, of the same C R dimension as M , defined by (2.3.7)
Im
Z3 :
I N 1 12
,
Z2 :
X
for x C R. Hence, it is not minimal at any point. Also, M is not locally biholomorphically equivalent to a real plane, and is therefore not Levi flat. 2.4. N o r m a l f o r m f o r g e n e r i c s u b m a n i f o l d s . In this section, we shall describe a convenient normal form for generic real analytic submanifolds. Let M C C N be a connected such submanifold. We write rn = C R codim ( M ) ( = codim ( M ) ) and n = C R d i m ( M ) , so that N = n + r n . We let P0 be a p o i n t in M. T h e n there are holomorphic coordinates Z = (z, w) vanishing at P0, with z E C ~ and w E C m, such t h a t M is given near P0 = 0 by (2.4.1)
Im w = r
z, Re w),
where r X,s) is a Cm-valued holomorphic function, Nm-valued for X = 2 and s E R m, such t h a t
(2.4.2)
r
-- r
X,s) _= 0.
Such coordinates are called normal coordinates, and (2.4.1) is called normal f o r m for M ; note that all examples given so far have been in normal form. We shall sketch a proof of the existence of normal coordinates at the end of this section. We refer the reader to [CM] or [BJT] for a detailed proof of the existence. If M is real algebraic, then there are algebraic normal coordinates at P0, i.e. the change of coordinates is algebraic and the function ~(z, 2, s) is algebraic. This follows readily from the proof of the existence of normal coordinates; as we shall see, the proof is based on an application of the implicit function, which preserves algebraicity in view of L e m m a 2.1.2. By writing Re w = (w + t~)/2 and Im w = (w - w ) / 2 i , we can solve for w in (2.4.1) by the implicit function theorem. We find that M consists of the set of points (z, w) for which (2.4.3)
w = Q(z, 2, w),
43 where Q(z, x, T) is a Cm-valued holomorphic function. It is straightforward to check that (2.4.2) implies
O(z, O, v) =_ Q(O, x, v) - r.
(2.4.4)
Note that (2.4.3) is not a R'~-valued equation for M. However, there is an m x m matrix-valued function a(z, w, X, r) such that (2.4.5)
w - Q(z, 2, ~) = a(z, w, 2, e ) ( I m w - r
2, Re w)).
By complex conjugating (2.4.3), we can also describe M by the equation
(2.4.6)
~ = O(~, z, w);
we use here the notation (2.4.7)
h(Z) = h(Z)
for a holomorphic function h(Z). An explicit basis L1, ...,L,, for the CR vector fields on M near p0 can be given in normal coordinates as follows (2.4.8)
Lj = ~zj +
~)k,~(2, z , w )
,
j = 1 , . . . ,n,
k=l
where we use the notation
(2.4.9)
Ok,~ (7, z, w) = ~ ( ~ ,
z, w).
Observe that the vector fields L1,..., Ln all commute. We conclude this section by sketching a proof of the existence of normal coordinates. We assume that M is given by (2.2.1) near P0. We may assume, by applying an affine change of coordinates in C N, that p0 = 0 and that
(2.4.10)
i
-
dpj(0, 0) = ~(dZ~+ i - dZn+j).
We write z = (Z1,..., Z,,), w = (Zn+l,..., Z~+m), and Z = (z, w). If we also let X E C n and r E C m, then equation (2.4.10) implies that we may solve (2.4.11)
p((z, w), (X, 7)) = 0
for r = S(X, z, w) near the point (z, w, X, 7) -- (0, 0, 0, 0). The function S(X, z, w) so obtained is holomorphic near (0, 0, 0). It follows that M near 0 consists of those points (z, w) for which (2.4.12)
~ = S(2, z, w).
44
If N = 1 and m = 1 (i.e. M is a real analytic curve in the complex plane), then M is defined by @ = S(w), where S(w) is a holomorphic function--the function S(w) is called the Schwarz function of the real analytic curve M and is a useful tool in the theory of one complex variable (see e.g. [D], [Sh]). Coordinates (z, w), vanishing at p0, such that the (2n + rn)-plane (2.4.13)
{(z,w) e C n x Cm: Im wl . . . .
Im Wm= 0}
is tangent to M are normal if and only if p((z, w), (0, w)) - 0, as is straighforward to verify from (2.4.1) and (2.4.2). To find normal coordinates (z', w'), we therefore look for a change of coordinates of the form (2.4.14)
(z,w) = ( z ' , w ' + h(z',w')),
where h(0, 0) and dh(0, 0) both vanish, such that
(2.4.15)
~' + ~(0, ~') - s(O,z',~' + h(z',w')) = 0;
for brevity, we shall omit the ' on the coordinates in what follows. In particular, the function h must satisfy (2.4.16)
w + f~(O,w) - S(O,O,w + h(O,w)) - O.
Now, let us agree to look for h(z, w) such that (2.4.17)
h(0, w)
=
-h(O,w).
Then h(0, w) is uniquely determined as the solution A = h(0, w) near (w, A) = (0, 0) of (2.4.18)
w - A - S(0, 0, w + A) = 0.
That (2.4.18) is uniquely solvable at (0, 0) follows from the implicit function theorem, because Sw(0, 0,0) equals the identity matrix (which is the reason we are looking for h(z,w) satifying (2.4.17) rather than ]~(0, w) = h(O,w)) as is easy to verify from (2.4.10)and the definition of S(X, z, w). We need to check that the solution h(0, w) so obtained actually satifies (2.4.17). To this end, we note from the definition of S(X, z, w) and the reality of p(Z, 2) that (2.4.19)
S(X, z, S(z, X, ~)) - r.
The function Iz(0, w) satisfies, by the definition of h(0, w), (2.4.20)
w - h(0, w) -= S(0, 0, w + h(0, w)).
Hence, equation (2.4.19) implies that (2.4.21)
S(0, 0, w
-
h(0, w)) - w + h(0, w),
45 and hence (2.4.17) holds, since h(0, w) is the unique solution of (2.4.18). We find the function h(z, w) as the solution ~ = h(z, w) of the equation
(2.4.22)
w + / 4 0 , w)
-
s(0, z, m + ~) =
o.
It is easy to check from the definition of S(X, z, w) and (2.4.10) t h a t (2.4.22) can indeed be solved and the solution ~ = h(z, w) satisfies (2.4.15) as well as h(0, 0) = 0, dh(O, O) = O. This proves the existence of normal coordinates for M . It is easy to see t h a t normal coordinates are not unique, although the change of coordinates we found above was uniquely determined by the special conditions we imposed on it. To this end, assume that ( z , w ) are normal coordinates for M . If f(z') is any C " - v a l u e d function with f ( 0 ) = 0, dr(O) = 0 and g(w') is any Cm-valued function with g(0) = 0, dg(O) -- 0, and g(w') = ~(w'), then it is straightforward to check t h a t the coordinates (z I, w I) defined by
(z, w) = (z' + f(z'), w + g(w'))
(2.4.23) are also normal.
2.5. The complexification of a real analytic s u b m a n i f o l d . Let, as above, M C C N denote a generic real analytic submanifold of codimension ra, and let P0 E M . E m b e d C N in C 2N -~ C N x C N as the t o t a l l y real plane {(Z, {) C C 2N : Z = ~}. T h e n there is a complex submanifold 34 C C 2N of (complex) codimension m near (P0, i60) such t h a t its restriction to the totally real plane C N equals M. We can define 34 as follows. Let p(Z, Z) = ( p l ( Z , Z), ..., pro(Z, 2 ) ) be local defining functions for M near P0 as in (2.2.1). We define 34 C C 2N by
(2.5.1)
34 = {(z, r 9 c~N: p(z, r = 0}.
If we choose normal coordinates Z = (z, w) vanishing at P0 and write ~ = (X, r ) , with X E C n and r E C m, then we can describe 34 near (P0,/%) = (0,0) as the graph (2.5.2)
w =
Q(z, x, r )
or
T = (~(x, z, w).
T h e complex manifold 34 is called the complexification of M. Moreover, M embedded as a submanifold of the totally real plane C N is a m a x i m a l l y real submanifold of 34. Thus, if h(Z, Z) = 0 is a real analytic equation t h a t holds for Z E M , then h(Z, ~) = 0 is a holomorphic equation that holds for (Z, ~) E 34. The complexification 34 will be used in the definition of the Segre sets of M. 3. HOLOMORPHIC NONDEGENERACY~ FINITE NONDEGENERACY, AND REFLECTION IDENTITIES FOR HOLOMORPHIC MAPPINGS 3.1. H o l o m o r p h i c nondegeneracy of real analytic C R s u b m a n i f o l d s . In the following two sections, we shall present some fairly recent notions of nondegeneracy of real analytic submanifolds in C N. By a holomorphic vector field X in some
46 neighborhood U of a point p E C N, we mean a holomorphic section of T(I'~ Z are local coordinates in U, this means
(3.1.1)
If
N 0 x -: Z a,(Z) , j+X OZj
where the coefficients aj(Z) are holomorphic functions in U. If a vector field is tangent to a submanifold M, we can take its restriction to M. We say that the restriction is trivia] if it is the trivial vector field on M. If a tangent vector field is given by (3.1.1), its restriction is trivial if and only if all the coefficients aj(Z) vanish on M.
Definition 3.1.2. A submanifold M C C N is said to be holomorphically degenerate at Po E M if there is a holomorphic vector field tangent to M near po such that its restriction to M is non-trivial. M is said to be holomorphicalIy nondegenerate at P0 if it is not holomorphically degenerate at that point. Holomorphic nondegeneracy, for real hypersurfaces, was introduced by Stanton in [St1] in which the connection with the infinitesimal CR automorphisms was investigated (see also [BER2] for higher codimensional results in this direction). The property of being holomorphically nondegenerate propagates along connected, real analytic CR submanifolds in the following sense.
Proposition 3.1.3 ( [ B E R 1 ] ) . Let M be a connected real analytic CR aubmanifold of C N, and let Pl,P2 E M. Then M is holomorphically degenerate at Pl if and only if it is holomorphically degenerate at p2, R e m a r k . The conclusion of Proposition 3.1.3 fails in general if we do not require that the connected real analytic submanifold is CR, An example showing this is given in [BER1, Remark 1.4.2].
Proof of Proposition 3.1.3. Since~ as observed in section 2.2, every CR submanifold in C N is a generic submanifold of some complex submanifold of C N, the proof can be reduced to that for a generic submunifold. Thus, we shall complete the proof under the assumption that M is a generic real analytic submanifold. We start with an arbitrary point P0 E M and we choose normal coordinates (z, w) vanishing at P0. We assume that M is given by (2.4.3), or equivalently by (2.4.6), for (z, w) near 0. We write
(3.1.4)
-Q(x, z, w) = E q,(z, w)x ~ o~
for Izh IxI, Iwl < 5. We shall assume that 5 is chosen sufficiently small so that the right hand side of (3.1.4) is absolutely convergent. Here q~ is a holomorphic function defined for [z[, [w[ < 5 valued in C m.
Assertion 3.1.5 (essentially [BR3]). Let ( z l , w 1) E M, with [z][, [w 1] < 5. If X is a germ at ( z l , w 1) of a holomorphic vector field in C N, then X i3 tangent to
47
M if a~d only if~ for (3.1.6)
(z,w)
aj(z,w)
X =
in 6~neighborhood o f ( z l , w l ) ,
aj(z,w)q~,~j(z,w) = 0,
and
j=l
Va e Z~_,
j=l
where the coefficients aj are holomorphic :in a neighborhood of (z 1, wl); here, the qa,z~ are the derivatives with respect to zj of the qa given by (3.1.4). Let us conclude the proof of Proposition 3.1.3, assuming that Assertion 3.1.5 has been proved. It follows easily by linear algebra from (3.1.6) that if M is holomorphically degenerate at a point (z 1, w 1) as above, then it is holomorphically degenerate at any point (z, w) in the local chart U of normal coordinates. (Apply Cramer's rule in a suitable vector space over the meromorphic functions in U; see e.g. the proof of [BR3, Lemma 4.9]). Proposition 3.1.3 now follows by the existence of normal coordinates at every point and the connectedness of M. Thus, the proof of Proposition 3.1.3 is complete as soon as Assertion 3.1.5 is proved.
Proof of Assertion 3.1.5. Assume that X = ~-'aj(z,w)w-- +
(3.1.7)
~(]ZJj=I
bk(z,w) k=l
, k
defined near (zl,wl), is tangent to M. This means that o
(3.1.8)
=x(~-O(z,z,w)) = -- ~ aj(z,w)Qzi (5, z,w) -- ~ j=l
bk(z,W)Qwk(5, Z,W),
k=l
for all (z,w) C M near ( z l , w l ) , i.e. for all solutions of t~ - Q(2, z,w) -- 0. As we remarked in section 2.5, the equation (3.1.8) continues to hold if we consider (z, w, 5, t~) as independent variables (z, w, u T), as long as r = Q(X, z, w) (i.e. for (z, w, X, T) E M ) . Thus, we have (3.1.9) j=l
aj(z, W)Qzj (X, z, w) + Z ' , bk(z, W)Qw~ (X, z, w) ----0, k=:
for all r = Q(X, z,w). However, since the left side of (3.1.9) is independent of T, (3.1.9) holds identically for all (z, w, X) near 0. Also, observe from the normal form (see (2.4.4)) that (3.1.10)
O~j(O,z,w)-O
,
~)~(O,z,w)=vk,
where vk is the m-dimensional vector with k:th component 1 and all other components 0. Taking X = 0 in (3.1.9), we conclude that bk(z,w) -- 0 for every k = 1, ..., rn. The proof of Assertion 3.1.5 is now completed by using bk(z, w) -- 0 in (3.1.9) and expanding Qzj (X, z, w)in accordance with (3.1.4). As remarked above, this also concludes the proof of Proposition 3.1.3. [] In view of Proposition 3.1.3, we shall say that a connected CR submanifold M is holomorphically nondegenerate if it is so at one (and hence at every) point.
48 3.2. F i n i t e n o n d e g e n e r a c y o f real a n a l y t i c C R submanifolds. We shall connect holomorphic nondegeneracy with a more computational form of nondegeneracy. Let M C C N be a generic real analytic submanifold with P0 E M. Assume that M is defined near P0 by (2.2.1). We denote by L1,...,Ln a basis for the CR vector fields near P0, and by pj_,z(Z, 2) the gradient Opj/OZ of the j : t h component of the defining function p(Z, Z). We use standard multi-index notation, i.e. L ~ denotes L~ 1...L~- for any multi-index c~.
Definition 3.2.1. M is ~aid to be finitely nondegenerate at Po if there exists an integer k such that (3.2.2)
spas {L~pj, z(po,Po): j = 1 , . . . , m , lal _< k} = C N.
If ko denotes the smallest integer for which (3.2.2) holds, then we also say that M is ko-nondegenerate at Po. It is straightforward to check that Definition 3.2.1 is independent of the defining equations p(Z, 2) = 0, the choice of basis L1,...,L~, and the coordinates Z. If M is finitely nondegenerate at p0, then the number ]Co in Definition 3.2.1 is a biholomorphic invariant. (This invariant was first introduced, in the hypersurface case, by Baouendi-Huang-Rothschild [BHR].) It is easy to see that a hypersurface M is l-nondegenerate at a point if and only if it is Levi nondegenerate at that point. There are examples of real analytic hypersurfaces which are not Levi nondegenerate at any point, but e.g. 2-nondegenerate at every point. (Consider e.g. the hypersurface in C 3 defined as the set of regular points of (Re Z1) 3 + (Re Z2) 3 + (Re Z3) 3 = 0). Also, note that a real submanifold M is 0-nondegenerate at a point P0 if and only if T~o(M ) = {0}. Since, for a real analytic CR submanifold M C C g at a point P0 E M, there is a unique (germ at P0 of a) complex submanifold X containing (the germ at P0 of) M such that M is generic in X, we can define finite nondegeneracy and k0nondegeneracy for a non-generic CR submanifold by considering it as a generic submanifold of X. We shall, for convenience, consider only generic submanifolds in what follows. Finite nondegeneracy is related to holomorphic nondegeneracy in the following way.
Proposition 3.2.2 ( [ B E R 1 ] ) . Let M be a connected real analytic generic submanifold of codimension rn in C N. Then the following conditions are equivalent. (i) M is holomorphically nondegenerate. (ii) There exists Pl E M and k > 0 such that M i8 k-nondegenerate at Pl. (iii) There exists V, a proper real analytic subset of M and an integer ~ = ~(M), 1 < g(M) < N - m, such that M is g-nondegenerate at every p E M \ V . We shall call the number g(M) given in (iii) above the Levi number of M. Proof. It is clear that (iii) implies (ii). We shall now prove that (ii) implies (i). Assume that M is k-nondegenerate at Pl. We take normal coordinates (z, w) vanishing at Pl, so that M is given by (2.4.3) (or (2.4.6)) near (z, w) = (0, 0). We take the vector fields in (2.4.8) as a basis of CR vector fields on M near Pl = 0. We
49 denote by Via(Z, 2 ) the vectors L~pj,z(Z, 2). In normal coordinates Z -- (z, w), we obtain (3.2.3)
Vj,(Z, Z) = -Oj,x,Z(-5, z, w).
The hypothesis (ii) implies that the vectors Vj~(0,0), j = 1 , . . . , m and laI < k, span C g. By using the normal form, we deduce that the vectors qjc,,z(O, 0), j = 1 , . . . , m and ]a] _< k, span cn; here, the functions qjc,(z, w) are the components of the vector q~(z, w) defined in (3.1.4). This implies, by linear algebra, that the only functions aj(z, w) that satisfy (3.1.6) in a neighborhood of 0 are the ones that vanish identically. Hence, M is not holomorphically degenerate at 0, proving (i). To show that (i) implies (iii), we shall need the following two lemmas, whose proofs are elementary and left to the reader. 3.2.4. Let f l ( x ) , . . . ,fm(X) be m holomorphic functions defined in an open set ~ in C p, valued in C g and generically linearly independent in ~. If the Oc~fj(x), j = 1,... ,m and a 9 Z~, span C g generically in fl, then the O~fj(x) (j = 1,... , m, [hi < N - m) also span C N generically in ft. Lemma
L e m m a 3.2.5. Let (z, w) be normal coordinates for M as above, and let h(x, z, w)
be a holomorphic function in 2n + m variables defined in a connected neighborhood in C 2n+m of z = z l , w = w l , x =-51 , with ( z l , w 1) 9 M. If h(-5, z,w) -- 0 for (z,w) 9 M, then h - O. To prove (i) implies (iii), we again take (z, w) to be normal coordinates around some point P0 9 M. By assumption (i) and (3.1.6), it follows that the qj~,z(z,w), j = 1 , . . . , m and all a, span C '~ generically. Equivalently, by the normality of the coordinates, we obtain that the -Qj,x,z(O,z,w) span C g generically. We claim that the "Qj,xaZ(-5, Z, W) span C N generically for (z, w) 9 M. Indeed, if the Qj,x~Z(-5, z,w) do not span at any point (z,w) 9 M, then all N x N determinants A(2, z, w) extracted from the components of these vectors vanish identically on M and hence, by L e m m a 3.2.5, A(X , z, w) ----0 in C 2n+m. In particular, A(0, z, w) -- 0, which would contradict the fact that the -Qj,x~z(0, z, w) span C N generically. This proves the claim. Now choose (z~ ~ 9 M so that A(O,z~ ~ r 0 for some determinant A as above. We apply L e m m a 3.2.3 with f j ( x ) = -Qj,z(X,z~176 = 1 , . . . , m , to conclude that there exists g < N - m such that in the local chart of Z = (z, w), the V j , ( Z , Z ) , for j = 1 , . . . m and ]a] _< g, span C N generically for Z 9 M. (The set where they do not span is defined by the vanishing of determinants and is thus a real analytic set). Since this property is independent of the choice of local coordinates, condition (iii) follows from the connectedness of M. This completes the proof of the equivalence of (i), (ii) and (iii). [] Let us record here for future reference the fact that, in normal coordinates, k-nondegeneracy at P0 = 0 is equivalent to (3.2.6)
span{Qj,xoZ(0,0,0): j = 1, ...,m, ]hi _< k} = C N.
Since Q(x, 0, w) = Q(0, z, w) - w, this in turn is equivalent to (3.2.7)
span {Qj,x~z(O, O, O) : j = 1, ...,m, [a] < k} = C ' .
50 3.3. Reflection identities for holomorphic m a p p i n g s . In this section, we shall deduce an important consequence of finite nondegeneracy. Let M, M r C C N be real analytic submanifolds, and let p0 E M, p~ E M ' . Suppose that H ( Z ) is a germ at p0 of a holomorphic mapping C N --~ C N such that H ( M ) C M ' and H(po) = P'o. We shall assume here that M is k0-nondegenerate at P0 and that H is invertible at P0. Denote by M C C 2N the complexification of M as described in section 2.5. We have the following "reflection identity" for the mapping H.
Proposition 3.3.X ( [ B E R 1 ] a n d [ B E R 2 ] ) .
Let M , M ' C C N be generic real analytic submanifolds, and let Po E M , P'o E M r. Assume that M is ko-nondegenerate at Po. Then there are C N - v a l u e d functions if2~, for all multi-indices 7, holomorphic in all of their arguments, with the following property. If H i~ a germ at Po of a biholomorphism o f C N such that H ( M ) C M ' and H(po) = P~o, then
(3.3.2)
(
Ol'~lH ~ ("Z") . q,~ . Z, .
0,o, (~), ) ,
r .H(~), . . , ~.
~heTe I"I <- k0 + t~t, for all points (Z, r E •
near (P0,P0). MoTeover, i / M and
M r are real algebraic then each q2~ is algebraic. Proof. Let (z, w) E C n • C m be normal coordinates (algebraic, if M is algebraic) for M vanishing at P0, i.e. M is defined near p0 by (2.4.3) or (2.4.6), and similarly for the target M ' (denoting the normal coordinates for M ' by (z', w'), the function defining M ' by Q', etc.). We write the mapping H as H = (f, g), where f ( z , w) E C n and g(z, w) E C m. Since H ( M ) C M r, .~(2, ~) = Q'(f(2, ~), f ( z , w), g(z, w))
(3.3.3)
holds for points (z, w) E M near P0 = 0. By complexifying (cf. section 2.5), we obtain .~()/, r) = Qr(f(X, r), f( z, w), g( z, w)),
(3.3.4)
for all (z, w, X, r) E A/l near 0. We define the following holomorphie vector fields Lj in C 2N, all tangent to A4 (and corresponding to the CR vector fields of M), (3.3.5)
Lj =
0_0 + Qk,• OXj k=l
w 0)-~rk , j = 1,... ,n.
We shall also need the following vector fields tangent to M
LJ = Ozj +
Qk,~j (z,)c,
,
j = 1,..., n,
k=l
(3.3.6)
Tj = Ow----~+
~)k,~j (X,
,
j = 1, ..., m,
k=l
Vj=Lj-~-~Qk,~i(z,x,v)7-k,
j=l,...,n.
k=l
Note that the coefficients of all the vector fields given by (3.3.5) and (3.3.6) are algebraic functions of (z, w, X, T) when M is real algebraic.
51 A s s e r t i o n 3.3.7. For all (z, w, X, 7-) G .M near 0 and all multi-indices 7 = (3", 7"), We have
(3.3.8)
-"-r . . . . OZ
(z,w)
V'~3T'~2Ef'~h(X,r),
O W "r
V
T. s
gZ(X,T) .... ),
""'
wherej, k = 1,...,n, l = 1,...,m, ]a1[,[/31[ _< k0, ]a2[,I/32] _< ]7"], ]a3[,[Z31 < [7'1, and the A~ are holomorphic functions of their arguments. Moreover, if M' is real algebraic, then the functions A~ are algebraic. Proof of Assertion 3.3.7. We apply the operators /~j to the identity (3.3.4), and use the fact that the matrix s at (z, w, X, r) = (0, 0, 0, 0) is invertible (which in turn follows from the fact that H is a biholomorphism at P0 = 0) to deduce that there are functions Fj (algebraic if M ' is real algebraic) such that, for points on A4 n e a r 0, (3.3.9)
- ' (f,- f, g) = F j ( s Qxj
s
we use here the convention that f = f ( z , w ) , f = f(X,v), etc. We repeat this procedure, using in the next step (3.3.9) instead of (3.3.4) and so on. We obtain, for every multi-index a, (3.3.10)
Q'xo(f , f ,g) = F,~(...,s f , ..., s
...),
where 1/31,171 _< I~1. Since H is a biholomorphism at P0, it follows that M ' is k0-nondegenerate at p~. Hence (see section 3.2), we have (3.3.11)
-' span{Qx-z(O,O,O): I~t <_ k0} = c ~.
We deduce, by applying the implicit function theorem to a suitable subcollection of the equations (3.3.10), that, for all (z, w, X, r) E .s near the origin, (3.3.12)
f j ( z , w ) = Oj(...,s163
,
j = 1,...,n,
where k = 1, ...,n, l = 1, ...,m, lal, 1/31 < k0, and the Oj are holomorphic (algebraic if M r is real algebraic) functions of their arguments (cf. e.g. [BR4, Lemma 2.3]). Now, since f(z, w) is a function of (z, w) only, we have, for any multi-index 7 =
(7 r, 7" ), (3.3.13)
, OI-dfj V ~ T ~ f ( z , w ) - _~ ~,o ./,(z,w). ,,
OZ
(]W
The assertion follows if we apply VT'T ~'' to (3.3.12), which we can do since the Vj and 7}/ are tangent to Ad. [] We now proceed with the proof of Proposition 3.3.1. As in (3.3.4), using (2.4.3) instead of (2.4.6) to define M r, we have (3.3.14)
gt( z, w) = Q't(f(z, w), f ( x , "r), .q(x, r))
52 for (z, w, X, r ) E M and l = 1, ..., m. If we apply
V'r'T ~'' to this equation we obtain
oh'lgt ~Z
7,,,(7W 7,,(z, w) =
(3.3.15) ...,
01~1+1~1 f
~
7-
~_
"~ /
where j = 1,...,n, k,l = 1,...,m, lal],lfllh[#ll < ICI, 1~1, l~21,1~21 ~ WI, and where the ~7 are holomorphic (algebraic if M ~ is real algebraic) functions of their arguments. Using (3.3.8), we obtain (3.3.16) ~z 7- ~2 f_.~a ol lg;
Oz'r Ow'r"
where j = 1,...,n, k,l = 1,...,rn, [a'[,[~ll, l~a[ < k0, I~zl,19~l,l~l _< I'/'l, and i~1, i/~3l, 1~31 < 17'1. The E~ are holomorphic (algebraic if M ' is real algebraic) functions of their arguments. By using (3.3.5) and (3.3.6), Proposition 3.3.1 follows from (3.3.8) and (3.3.16). [] Let us explain why we choose to call (3.3.2) a reflection identity. For simplicity, we shall confine this discussion to the case where M and M ~ are real analytic hypersurfaees. If we restrict ~ to Z, then (3.3.2) with 7 = 0 says that on M each component of H coincides with a holomorphic function of Z, Z, and derivatives of H ( Z ) . By using similar arguments to those in the proof of Proposition 3.3.1, one can show that a corresponding identity holds for all smooth (e.g. C ~ ) CR mappings F of M into M t, i.e. smooth mappings of M into M ~ whose components are CR functions. If the components of F extend holomorphically to one side of M then, by "flattening" M using a change of coordinates that is only holomorphic in the transversal direction, one can use (3.3.2) to define a holomorphic m a p p i n g on the other side of M such that this reflected mapping matches the values of F on M. By using the edge-of-the-wedge theorem, one can then show that F in fact extends as a holomorphic mapping in a full neighborhood of M near P0. We refer reader to e.g. [BJT] and [BR1] for details on this. In this paper, we shall use the identity (3.3.2) in a different way. The identity will allow us to "propagate" certain properties (e.g. algebraieity) of holomorphic mappings from a single point to a larger set along the Segre set, which we shall introduce in section 4.1. As a motivation for this idea, note the following: since .M is defined by w : Q(z, x, r) and since Q(z, 0, r) -- r, the point (z, w, X, T) = (z, 0, 0, 0) is in M for all z E C'* near 0. Substituting this point in (3.3.2), we obtain (3.3.17)
OITIH ~ (z,0)=~
(
(z,0),(0,0),H(0,0),...,
OIC'lH ) O-U ( 0 , 0 ) , . . . .
In particular, H(z, 0) is completely determined by the values of its derivatives up to order k0 at 0. Moreover, if M and M t are real algebraic, then it follows from (3.3.17) that O~H(z,O)is algebraic. This information about O~H(z, 0) translates into information about O~H(x, 0) that we can substitute back in the right side of
53 (3.3.2). Under suitable circumstances, this leads to information on O~H on a larger set. The main difficulty is to describe the largest set on which we, by iterating this procedure, obtain information about H. This set turns out to be the largest Segre set. 4. THE
SEGRE SETS
4.1. The Segre sets o f a r e a l a n a l y t i c C R s u b m a n i f o l d . Let M denote a generic real analytic submanifold in C N, and let p0 C M. We assume that M is defined near P0 by (2.2.1). Embed C u in C ~N = C zN x C~v as the real plane { ( Z , ( ) C c 2 N : ( = Z}, and let A,4 C C ~N be the complexification of M (see (2.5.1)). Let us denote by p r z and prr the projections of C 2N onto C zN and C ~ , respectively. The natural anti-holomorphic involution ~ in C 2N defined by (4.1.1)
U(Z, 4) = (~, Z)
leaves the plane {(Z, (): r = Z} invariant. This involution induces the usual antiholomorphie involution in C N by (4.1.2)
C N ~ Z ~ prr
= 2 E C N.
Given a set S in C N we denote by *S the set in C~v defined by (4.1.3)
*S ----p r r
= {~: ~ C S}.
By a slight abuse of notation, we use the same notation for the corresponding transformation taking sets in C ~ to sets in C zN. Note that if X is a complex analytic set defined near Z ~ by hi(Z) . . . . . hk(Z) = 0 in some domain f~ C C zN, then *X is the complex analytic set in *fl C C~v defined near C~ = 2 ~ by/~a(~) = . . . . hk(~) = 0. Thus, the transformation * also preserves algebraicity. Observe that 3/1 is invariant under the involution It defined in (4.1.1). Indeed all the defining functions p(Z, 2) for M are reai-valued, which implies that the holomorphic extensions p(Z, ~) satisfy (4.1.4)
fi(Z, ~) = p(~, Z).
Hence, given (Z,~) E C 2N we have p(~(Z, ~)) = p(G Z) = f(C,, Z) = p(Z, ~). It follows that ~(Z, ~) E M if and only if (Z, r E A4. We now come to the main construction in this section. We associate to M at P0 a sequence of germs of sets No,Na,..., Njo at Po in CN--called the Segre set~ of M at p0--defined as follows. Define No = {P0} and define the subsequent sets inductively (the number j0 will be defined later) by (4.1.5)
NiT 1 = p r z (.]~Npr~l(*ij))
= p r z (.hd N +prz1 ( N j)) 9
Here, and in what follows, we abuse the notation slightly by identifying a germ Nj with some representative of it. These sets are, by definition, invariantly defined
54 and they arise naturally in the study of mappings between submanifolds (see e.g. section 5.2). Let the defining functions p and the holomorphic coordinates Z be as in section 2.1; for notational convenience, we shall assume that the coordinates Z vanish at P0, i.e. P0 = 0. The Segre sets Nj can then be described as follows. For odd j = 2k + 1 (k = 0, 1, ...), we have
(4.1.6)
N2k+l = {Z: ~Z1,...,zk,(1,...,(k: p(Z,(k) z p(zk ~ k-l) . . . . . fl(zl, 0) = O, p(Z k, ~k) = p(Zk-1, ~k-1) . . . . . fl(Z 1, ~1) = 0};
observe that for k = 0 we have N1 = {Z: p(Z,O) = 0}.
(4.1.7)
For even j = 2k (k = 1,2, ...), we have
(4.1.8)
N2k = {Z : ~ZI,..., Z k-l, ~1 "", (k : p(Z, ~k) = p ( z k - 1 ~-k-1) . . . . . /)(Z 1' ~1) = O, p(Zk-l,r k) : p(zk-2,~ k-l) . . . . . p(O,~ 1) = 0 } .
For k = 1, we have (4.1.9)
d~2
=
{Z: ~ 1 : p ( z , ~ l ) =
0, p(0,(1) = 0}.
From (4.1.6) and (4.1.8) it is easy to deduce the inclusions (4.1.10)
No c N~ c ... c N~ c
. . .
When m = 1 the set N1 is the so-called Segre surface through 0 as introduced by Segre [S], and used by Webster [W1], Diederich-Webster [DW], Diederich-Fornaess [DF], Chern-Ji [C J], and others. The set N2 is the union of Segre surfaces through points ~1 such that ~1 belongs to the Segre surface through 0. Subsequent Nj's can be described similarily as unions of Segre surfaces. In order to simplify the calculations, it is convenient to use normal coordinates Z = (z, w) for M as in section 2.4. Recall that M is assumed to be generic and of codimension m; we write N = n + m. Assume that M is given by (2.4.3). We shall write (4.1.11)
Q ( z , x , ~ ) = ~ + q(z, x, ~),
where (4.1.12)
q(z,O,r) =_q(O,x,r) =- O.
In C ~-N, we choose coordinates (Z, () with Z = (z, w) and ~ = (X, T), where z, X e C n and w, r C C "~. Thus, as noted in section 2.5, the complex manifold A// is defined by either of the equations (4.1.13)
w = Q ( z , x , v ) or 7 = Q ( x , z , w ) .
55 In n o r m a l coordinates, we find t h a t in the expression (4.1.6) for N2k+l w e can solve recursively for w 1, 7.1, w 2, 7.2,..., w k, rk and p a r a m e t r i z e N2k+l by (4.1.14)
C (2k+l)n 2 (z,z 1, ...,zk,x 1, ...,X k) = A ~-~ (z, v2k+l(A)) E (]N,
where
(4.1.15)
V2k+l(A) = T k + q(z, xk,7.k),
and recursively
(4.1.16)
~ 7.1-1 +q(zl,xl-1 7.t=Wl+~(Xt, Zl,Wl ) with w l = [ 0 , / = 1
7.1-1), l > 2
for l = 1,2, ..., k; for k = 0, we have v 1 - 0. Similarily, we can p a r a m e t r i z e N2k by (4.1.17)
C 2kn
~ (Z, Z 1, ..., Z k - 1 , ~ 1 , "", x k ) =
A ~ (z, v2k(A)) E C N,
where (4.1.18)
v2k(A) = T k + q(z, X k, 7.k),
and recursively (4.1.19)
7.1+1 = w t + q(xZ+l, z l, w t) with w ~ = 7.z + q(z z, Xl 7.1),
for l = 1, ..., k - 1 and 7"1 = 0. Define dj to be the m a x i m a l rank of the m a p p i n g (4.1.14) or (4.1.17) (depending on whether j is odd or even) near 0 C C j ' . It is easy to see t h a t do = 0 and dl = n. In view of (4.1.10), we have do < dl _< d2 _< da _< . . . . We define the n u m b e r J0 _> 1 to be the greatest integer such t h a t we have strict inequalities (4.1.20)
do < dl < ... < djo.
Clearly, j0 is a well defined finite number because, for all j , we have dj < N = n + m and djo _> n + J0 - 1 so t h a t we have j0 _< m + 1. The dj's stabilize for j _> J0, i.e. djo = dj0+l = dj0+2 . . . . . , by the definition of the Segre sets. So far we have only considered generic submanifolds. If M is a real analytic C R submanifold of C N, then M is generic as a submanifold of its intrinsic complexification A' (see section 2.2). If M is real algebraic then A' is complex algebraic. The Segre sets of M at a point P0 E M can be defined as subsets of C N by the process described at the beginning of this subsection (i.e. by (4.1.5)) just as for generic submanifolds or they can be defined as subsets of A" by identifying X near P0 with (C~" and considering M as a generic submanifold of C ~:. It is an easy exercise (left to the reader) to show t h a t these definitions are equivalent (i.e. the l a t t e r sets are equal to the former when viewed as subsets of c N ) . The m a i n result concerning the Segre sets is the following. Let the H S r m a n d e r numbers, with multiplicity, be defined as in section 2.3.
56 T h e o r e m 4.1.21 ( B E R 1 ] ) . Let M be a real analytic CR submanifold in C N o f CR dimension n and of CR codimension rn and po E M. Assume that there are r (finite) HSrmander numbers of M at Po, counted with multiplicity. Then the following hold. (a) There is a holomorphic manifold X of (complex) dimension n + r through Po containing the maximal Segrc set Njo of M at Po (or, more precisely, every su~ciently small representative of it) such that Njo contains a relatively open subset of X . In particular, the generic dimension djo of Njo equals n Jvr,
(b) The intersection M N X is the CR orbit of the point Po in M. (c) If M is real algebraic then X is complex algebraic, i.e. X extends as an irreducible algebraic variety in C N. (d) There are holomorphic immersions Zo(to),Zl(tl) .... , Zjo(tjo ) defined near the origin,
C d~ 9 tj ~ Zj(tj) E C N,
(4.1.22)
and holomorphic maps So(tl),..., Sjo-1 (tjo), C dj ~ tj ~ sj_l(tj) E C ~i-1,
(4.1.23)
such that Zj(tj) has rank dj near the origin, Zj(tj) e Nj, and such that (Zj(tj), 2 j - l ( s j - l ( t j ) ) ) e M ,
(4.1.24)
for j = 1,...,j0. Moreover, if M is real algebraic, then all these maps are holomorphic algebraic. R e m a r k 4.1.25. The property (d) above is of technical importance as we shall see in the proof of Theorem 5.1.1 below. The mapping Zj(tj) parametrizes a piece of the Segre set Nj. Note that we do not claim that Zj(O) = 0, so the piece parametrized need not, and in general does not, contain 0. In particular, this theorem gives a new criterion for M to be of finite type (or minimal) at P0. The following is an immediate consequence of the theorem. C o r o l l a r y 4.1.26. Let M be a real analytic CR submanifold in C N of CR dimension n and of CR codimension m and Po E M. Then M is minimal at Po, if and only if ~he generic dimension djo of the maximal Segre set Njo of M at po is n + m. In particular, if M is generic, then M is minimal at Po if and only if djo = N. E x a m p l e 4.1.27. Let M C C a be the generic submanifold defined by I m w l = l z ] 2,
I m w 2 = l z ] 4.
Then M is of finite type at 0 with H6rmander numbers 2, 4. The Segre sets N1 and N2 at 0 are given by (4.1.28)
N1
= { ( Z , W l , W 2 ) : W 1 = O, W 2 =
0},
57
=
(4.1.29)
= 2izx,
= 2iz
x
, x e
C}.
Solving for X in (4.1.29) we obtain in this way (outside the plane {z = 0}) g 2 = { ( Z , W l , W 2 ) : w2 = - i w 2 1 / 2 } .
Using the definition (4.1.5), we obtain Y 3 ~- { ( Z , W l , W 2 ) : w2 = i W l ( W l / 2 - 2 z x ) ,
X E C}.
We have d3 = 3; N3 contains C 3 minus the planes {z = 0} and {wl = 0}. E x a m p l e 4.1.30. Consider M C C 3 defined by I m w l = [ z [ 2,
I m w 2 = R e w 2 [ z [ 4.
Here 2 is the only HSrmander number at the origin. Again, N1 is given by (4.1.28), and = {(z, wl,w ): z # 0,w2 = 0) u { 0 , 0 , 0 } . It is easy to see that subsequent Segre sets are equal to N2. Thus, N2 is the maximal Segre set of M at 0 , d2 = 2, and the intersection of (the closure of) N2 with M equals the CR orbit of 0. Let us also note that part (c) of Theorem 4.1.21 implies the following. C o r o l l a r y 4.1.31.
The CR orbits of a real algebraic CR submanifold are algebraic.
This corollary can be viewed as an "algebraic version" of Nagano's theorem for CR vector fields of real algebraic submanifolds. The theorem of Nagano ([N]) states that the integral manifolds of a system of vector fields, with real analytic coefficients, are real analytic. It follows that the CR orbits of a real analytic CR manifold M are real analytic submanifolds of M. However, in general the integral manifolds of a system of vector fields with real algebraic coefficients are not algebraic manifolds, as can be readily seen by examples. On the other hand, Corollary 4.1.31 implies that if, in addition, the system of vector fields comes from a CR structure embedded as a real algebraic submanifold in complex space then the integral manifolds are indeed algebraic. Before we prove Theorem 4.1.21 (in w we first discuss the homogeneous case because the proof of the theorem will essentially reduce to this case. Moreover, we shall only prove the theorem in the case where the CR dimension is 1. The general case is more technical, but the idea of the proof is the same. We refer the reader to [BER1] for the full proof of Theorem 4.1.21. 4.2. H o m o g e n e o u s s u b m a n i f o l d s o f C R d i m e n s i o n 1. Let #1 <- ... -< #N be N positive integers. For t > 0 and Z = ( Z I , . . . , Z N ) E C N, we let /~,Z = (tvlZ1,... ,t#~Zg). A polynomial P(Z,-Z) is weighted homogeneous of degree c with ~espect to the weights # 1 , . . . , #N if P(~tZ, ~ Z ) = tcp(z, Z) for t > 0.
58 In this section, we consider submanifolds M in C N, N = n + m, of the form
wl = wl + ql(z,~)
W j -~ W j + q j ( Z, Z, Wl ,
(4.2.1)
...,//3j-1)
M: Wr=~r+qr(z,2,~t,...,ffJr--1)
Wr+l
LUr+I
Wm ~ Win,
where 0 < r < m is an integer (r = 0 corresponds to the canonically flat submanifold), and each qj, for j = 1, ..., r, is a weighted homogeneous p o l y n o m i a l of degree m j . The weight of each zj is 1 and the weight of Wk, for k = 1, ...,r, is ink. Since the defining equations of M are polynomials, we can, and we will, consider the sets No, ..., Nj0 a t t a c h e d to M at 0 as globally defined subsets of C N. Each N j is contained in an irreducible complex algebraic variety of dimension dj (here, an algebraic variety of dimension N is the whole space CN). The l a t t e r follows e.g. from the p a r a m e t r i c definitions (4.1.14) and (4.1.17) of N j and the algebraic implicit function theorem ( L e m m a 2.1.2 (iii)). We let 7rj, for j = 2, ..., m + 1, be the projection 7rj : C ~+m ~-* C n+j-1 defined by
7rj(z, Wl, ...Wm) = (Z, Wl, ..., Wj--1).
(4.2.2)
We define M j C C "+j-1 to be 7ri(M ). By the form (4.2.1) of M , it follows t h a t each M j is the C R manifold of codimension j - 1 defined by the j - 1 first equations of (4.2.1). T h r o u g h o u t this section, we work under the assumption t h a t M satifies the following. C o n d i t i o n 4.2.3.
The CR manifold M j, for j = 2, ..., r W 1, is o/finite type at O.
We shall only consider the case where the C R dimension is 1, i.e. z E C. The m a i n technical result is the following. P r o p o s i t i o n 4 . 2 . 4 ( [ B E R 1 ] ) . Let M be of the form (4.2.1) with CR dimension n = 1 and assume that M satisfies Condition 4.2.3. Let N 0 , N ] , ...,Njo be the Segre sets of M at O, and let do,dl, ...,djo be their generic dimensions. Then jo = r + 1 and dj = j , for 0 < j < r + l. Furthermore, for each j = 0 , . . . , r + l , there is a proper complex algebraic variety Vj C C j such that Nj satisfies (4.2.5)
Nj N ((C j \ Yj) x C rn-j+l) =
{(Z, Wl,
...,Win)
e ((C j \ Yj) x C m-j+l) : w k
= fjk(z,w,,
...,wj-1),
k = j, . . . , m } ,
where each fjk, for k = j , . . . , r , is a (multi-valued) algebraic function with bjk holomorphic, disjoint branches outside Vj and where f jk -- 0 for k = r + 1, ..., m. Proof. Clearly, the first statement of the proposition follows from the last one. Thus, it suffices to prove that, for each j = 0, ..., r + 1, there are algebraic functions
59
fjk and a proper algebraic variety Vj such that (4.2.5) holds. The proof of this is by induction on j. Since No = {0} and N1 = {(z,w): w = 0}, (4.2.5) holds for j = 0 and 1 with V0 = V1 = !3. We assume that there are V0, ..., Vl-1 such that (4.2.5) holds for j = 0, ..., 1 - 1. By (4.1.5), we have (4.2.6)
Nt = {(z,w): 3(x,'r) 9 *Nt-1, (Z,W,X,T) 9 .It4}.
A s s e r t i o n 4.2.7. The set of points (z, wl,...,wt-1) 9 C t such that there exists (Wl,...,Wm) 9 C m-/+l and (X,T) 9 *(N/-1 C/(V/_ 1 • c m - l + 2 ) ) with the property
that (z, w, X, T) 9 3,1 is contained in a proper algebraic variety At C C t. Proof of Assertion ~.2.7. Let S be the set of points (z, wl, ..., wt-1) 9 C t described in the assertion. Then (z, wl, ...,wt-1) 9 C t is in S if (4.2.8)
T j - ~ w j + q j ( x , Z , Wl,...,Wj_I)
,
j = 1,...,/-- 1.
for some (X, T1,..., Tl-1) 9 *(Trl(Nt_l)A (V/_ 1 X C)). (Recall the two equivalent sets of defining equations, (4.1.13), for Ad. The operation * here is taken in C t, i.e. mapping sets in C (z,wl t t ..... w,_~)to C (• We claim that the set S is contained in a proper algebraic variety At C C t. To see this, note first that (4.2.5) (which, by the induction hypothesis, holds for N , - I ) implies that ~rt(Nz-x ) is contained in a proper irreducible algebraic variety in C t. Let PI(X, rl, ..., rt-2) be a (non-trivial) polynomial that vanishes on *Vl-1 C C 1-1, and let P2(X, rl, ...,rl_l) be a (non-trivial) irreducible polynomial that vanishes on *Trt(Nt-1). Thus, if (z, wl,...,wt-1) 9 S then there exists a X 9 C such that
(4.2.9) /51(X,Z, W1, ...,Wl--2) := P l ( ~ , w 1 + ql()(,z), ..., Wl_2 + ql-2()~,Z,Wl,..., Wl-3) ) = 0 /52(x,z, wl,...,wt_l)
:= P
(x, w l +
....
= o,
i.e. R(z, Wl,... , Wl_ 1) = 0 if we denote b y / ~ the resultant of/51 and t52 as polynomials in :g. The proof will be complete (with At = / ~ - 1 ( 0 ) ) if we can show that /) is not identically 0, i.e. /51 and/52 have no common factors (it is easy to see that neither/51 nor t52 is identically 0). Note that, for arbitrary rl,..., rt-1, we have (el.
(4.M3)) (4.2.10) /52(X,Z, T1 +ql(Z,)(),...,Tt_ 1 + q t _ I ( Z , X , T1,...,Tt_2)) = P2(X, T1,...,Tt_I). It follows from this that 152 is irreducible (since P2 is irreducible). Thus,/51 and t52 cannot have any common factors because/52 itself is the only non-trivial factor of /52 and, by the form (4.2.5) of Nz-1,/52 is not independent of wt-1. This completes the proof of Assertion 4.2.7. [] We proceed with the proof of Proposition 4.2.4. Let us denote by Bt C C I-1 the proper algebraic variety with the property that (z, wl,...,wl-2) E C t-1 \ Bt implies that the polynomial/51(X, z, wl, ..., wt-2) defined by (4.2.9), considered as a polynomial in X, has the maximal number of distinct roots. Let Ct C C t denote
60 the union of Al and Bi x C. For (z, wl, ..., w~-2) fixed, let f~(z, w~,..., wl-2) C C be the domain obtained by removing from C the roots in X of the polynomial equation (4.2.11)
Pl(X,z, Wl,...,Wl_2)
=
O.
In view of Assertion 4.2.7 and the inductive hypothesis that (4.2.5) holds for Nl-1, it follows from (4.2.6) that (4.2.12) Y~ n ((Cl \ Ci) • Cr"-l+1) = {(Z, Wl,...,Wrn) e ((C j \ e l ) x
cm-j+l)
:
3 x E ~'~(Z, Wl,...,Wl--2) C C , w k -~ g t k ( X , z , w , , ...,Wk--1), ]g = l -- 1,.., m},
where (4.2.13)
g,k(X, z, Wl,... , Wk-1) = J~-l,k(X, Wl -[- (~1()(', Z), ..., Wl_ 2 + ql-2(X, z, Wl, ... , w l - 3 ) ) ...[-q k ( z , X , W l + ql()(, z), ...,Wk-1 + (~k-l(X,Z,Wx, ..-, Wk-2)),
for k = l - 1,...,m. Note that each glk, for k = I - 1,...,r, algebraic function such that all branches are holomorphic in every point (X, z, w) considered in (4.2.12), and glk -- 0 for k = Now, suppose that gm-l(X, z, Wl, ..., wz-2) actually depends (4.2.14)
c3gt l--i
is a (multi-valued) a neighborhood of r + 1, ..., m. on X, i.e.
( X ' Z ' W l ' ' ' " W l - - 2 ) ~ 0"
0 W~_l) such that one branch g of gl,t-1 is holoThen, for each ( X ~ 1 7 6 0 morphic near (X ~ z ~ w ~ ..., w~ with (4.2.15)
00gx(X0 ~ z 0 ' ~i,..., 0 0 wl-2)
r
0
and (4.2.16)
w~_, g(x ~ z ~ w ~ ..., w~_2), =
we may apply the (algebraic) implicit function theorem and deduce that there is a holomorphic branch O(z, Wl,..., Wl-1 ) of an algebraic function near (z ~ w~ w~_ 1) such that
(4.2.17)
w t - 1 - g ( O ( z , w ~ , . . . , W ~ _ l ) , Z , W l , ..., w,-2) -= 0.
Since gl,t-1 is an algebraic function, which in particular means that any two choices 0 , wl_2) 0 of branches g at (possibly different) points (X ~ z ~ wl,... can be connected via a path in (X, z, Wl,..., wt-2) space avoiding the singularities of gl,l-1 and also avoiding the zeros of Ogt,l_l/Ox, it follows that any solution 0 of (4.2.17) near a point (z ~ wl, 0 ..., wl_l) 0 can be analytically continued to any other solution near a (possibly different) point. Thus, all solutions 0 are branches of the same algebraic
61
function, and we denote that algebraic function by 01. As a consequence, there is an irreducible polynomial R t ( X , z, wl, ..., wl-1) such that X = ~t(z, wl,..., wt-1) is its root. Let Dr C C l be the zero locus of the discriminant of Ri as a polynomial in X. Outside (Ci U Dr) • C m-t+l C C re+l, we can, by solving for X = ~t(z, Wl, ..., Wl--1) in the equation
(4.2.1s)
wl-1 = gt,r-l(x, z, Wl,..., w l - 2 ) ,
describe Nr as the (multi-sheeted) graph
(4.2.19)
Wk ---- frk(Z, Wl, ...,wl-1) : : glk(Ol(Z,Wl, . . . , w l - 1 ) , W l , ..., Wk--1),
for k = l, ..., m. Clearly, we have flk = 0 for k = r + 1, ..., m. By taking V/to be the union of Cl U Di and the proper algebraic variety consisting of points where any two distinct branches of flk coincide (for some k = l, ..., m), we have completed the proof of the inductive step for j = l under the assumption that g i j _ l ( X , z , w l , ..., wl-2) actually depends on X. Now, we complete the proof of the proposition by showing that Condition 4.2.3 forces (4.2.14) to hold as long as l - 1 < r. Assume, in order to reach a contradiction, that gl,l-l(X, z, wl, ..., w~-2) does not depend on X. It is easy to verify from the form (4.2.1) of M that the sets 7rk(Nj), for j = 0, ..., k, are the Segre sets of M k at 0. Let us denote these sets by N j ( M k ) . Now, note that if we pick (z o, wl,...,Wg_~ ) o o EMt then (4.2.20)
w0 +
z0), ..., w 0
+
z 0,
...,
=
...,
Thus, if we pick the point (z ~ w~ w~_l) E M z such that it is not on the algebraic variety Ct (which is possible since the generic real submanifold M l cannot be contained in a proper algebraic variety; Cl (3 M l is a proper real algebraic subset of M t) then, by construction of Cl, the point (4.2.21)
(5 ~
W I0 • q l- ( z
-0
0 2-~-ql_2(z -0 ,Z 0 ,Wl,... 0 ,Z 0 ),...,Wl_ ,w0 3)) = (~.0,~0,...,?.~0_2)
is not in *~'1(~-1). By the induction hypothesis, rt(Nl-1) = N I - I ( M l) consists of a b l - l j - l - s h e e t e d graph (each sheet, disjoint from the other, corresponds to a branch of ft-1 j - l ) above a neighborhood of the point (z ~ w~ ..., w~ Since gl,z-1 is assumed independent of X, we can, in view of (4.2.21), take X = 2 in the defining equation
(4.2.21)
wl-1 =- gt,t-l (X, z, wl , ..., wl-2 )
for N l ( M l ) , near the point (z ~ w ~ ..., w~ From the definition (4.2.13) of gl,z-1 and (4.2.21) it follows that N I ( M ~) also consists of a b-sheeted graph, with b _< bl-l,z-1, (each sheet corresponds to a choice of branch of ~ - l , t - 1 at the point 9", l-~)) above a neighborhood of the point (z ~ w ~ ..., w~ Because of the inclusion N~_I(M t) C Nl(Mt), we must have b = bl-l,l-1 and, moreover, for
62 k! each branch ff-l,l-1 there is possibly another branch fl--l,l--1 such that for every (z, Wl,..., wl-2) the following holds
(4.2.23) f L l , , _ l ( Z , Wl, ...,w,_2) = ~k_l 1,/_l(ff. , W 1 + q I ( Z , Z), ..., Wl-- 2 "~- (~/--2(L', Z, Wl, ... , Wl--3))
+ ql_l(Z, Z, Wl + (~I(Z, Z ) , - . . , Wl-- 2 ~C q/--2(Z, Z, WI, ... , W/--3) )S i n c e all the sheets of the graphs are disjoint, the mapping k -~ k t is a permutation. We average over k and k', restrict to points (z, wl, ..., Wl-2) C M t - l , and obtain, by (4.2.21) and (4.2.23), (4.2.24)
1
bl-lfl-I
f ik- l ' t - l ( Z ' W l ' " " W l - 2 )
b,_1,_1
'
-
1 bl-ll-1
bl-lfl-1
-,
k'=l
k=l
+ q l - l ( z , 2 , ~ l , .-., @-2). Let us denote by f the holomorphic function near (z ~ Wl~ ..., w~_2) defined by (4.2.25)
1 - - I-1 f ( z , w l , ...,Wl--2) -- bl-1
bl-l,t-1
'
ft_l,t_l(Z, Wl, ...,wt-2), k
E k=l
and by K C C l the CR manifold of CR dimension 1 defined near (z ~ , w ~ "", w 1--2, ~ f ( z ~ w ~ ..., w~ by (4.2.26) g :-~ {(Z, W l , . . . , W l _ l ) : (Z, Wl, ...,Wl_2) C M l-1 , Wl-1 -~ f ( z , w l , . . . , w l - 2 ) } . The equation (4.2.24) immediately implies that K C M ~. By Condition 4.2.3, M l is of finite type near 0. Note that, by the form (4.2.1) of M, the condition that M z is of finite type at a point is only a condition on (z, Wl, ..., wz-2) (i.e. not on wt-1). Thus, by picking the point (Z o , W lo, . . . , W~_2) ~ MZ-1 sufficiently close to 0 (which is possible since, as we mentioned above, Ct M M ~ is a proper real algebraic subset of M1), we reach the desired contradiction. This completes the proof of Proposition 4.2.4. []
4.3. P r o o f of Theorem 4.1.21 ( C R dimension 1 case). By the remarks preceding the theorem, we may assume that M is generic throughout this proof. Also, as we mentioned above, we shall only prove the theorem under the assumption that the C R dimension is 1. The difficulty (which is only technical) for higher CR dimensions is in proving the equivalent of Proposition 4.2.4. We start by proving (a). Since the Segre sets of M at P0 are invaxiantly defined, we m a y choose any holomorphic coordinates near P0. Let ml < ... < mr be the H5rmander numbers of M at P0. By [BR1, Theorem 2], there are holomorphic coordinates (z, w) E C • C'* (recall that the CR dimension of M is assumed to be 1) such that the equations of M near P0 are given by wj = ff~j + q j ( z , 2 , ffJl,...,wj-1) + R j ( z , 5 , ff~) , (4.3.1)
d wk=@k + E l=r+l
hkl(z,2, w)@z
,
j = 1,...,r
k = r + l,...,rn,
63 where, for j = 1, ..., r, qj(z, 5, wl,..., t~j--1) is weighted homogeneous of degree m j, R j ( z , ~, t~) is a real analytic function whose Taylor expansion at the origin consists of terms of weight at least mj-b 1, and the hkz are real analytic functions that vanish at the origin. Here, z is assigned the weight 1, wj the weight m j for j = 1, ..., r and weight m r + 1 for j = r + 1, ..., m. Moreover, the homogeneous manifold M ~ C C N defined by wj=t~j+qj(z,2,
(4.3.2)
wk = t~k
,
t~,...,ffJj_l)
,
j=
l,...,r
k = r + l,...,d
satisfies Condition 4.2.3. For e > 0, we introduce the scaled coordinates (5, t~) E C l+m defined by (4.3.3)
wj = w j ( C v ; e ) = e ljffJj
,
j = l,...,m,
where lj = m j for j = 1, ..., r and lk = m r + 1 for k = r + 1, ..., m. We write /zkl for the function hkl( Z, Z, W; ~) --~ l hkl( Z( Z ; ~), ~'(Z;
(4.3.4)
~), 7.~(W; ~)),
and similarly,
nj(5, 5, ~; e) - ~rnj1 +1 Rj(z(5; ~), ~(5; ~), e ( e ; ~)).
(4.3.5)
Note that both hkl(5, ~, t~; e) and/~j(5, ~, t~; e) are real analytic functions of (5, tS; e) in a neighborhood of (0, 0; 0). In the scaled coordinates, M is represented by the equations
{
ff~j = (vj + qj(s 3, ~vl,..., ~vj-1) + e/)j(5, ~, t~; e)
(4.3.6)
wk=t~k+e
~ /tkl(~,5, t~;e)t~l l=rq-1
,
,
j = 1, ..., r
k=r+l,...,m,
Now, let ~J(A; e) be the mapping C jn ~-* C m, described in section 4.1, such that the Segre set N j of M at P0 is parametrized by (4.3.7)
c~ ~ h ~ (5,,~(3,; ~)) ~ c N
in the scaled coordinates ( 5 , ~ ) (cf. (4.1.14)-(4.1.16) and (4.1.17)-(4.1.19) to see how the m a p (4.3.7) is obtained from the defining equations (4.3.6)). Note that oJ depends real analytically on the small parameter e. The generic dimension dj of the Segre set Nj is the generic rank of the mapping (4.3.7) with e r 0, and is in fact independent of e. By the real analytic dependence on e there is a neighborhood lr of e = 0 such that the generic rank of (4.3.7), for all e E I\{0}, is at least the generic rank of (4.3.7) with e = 0. For e = 0 the mappings (4.3.7) parametrize the Segre
64 sets N ~ of the homogeneous manifold M ~ defined by (4.3.2). By Proposition 4.2.4, applied to the Segre sets N ~ of M ~ at 0, we deduce that the generic dimension of the maximal Segre set of M ~ at 0 is r + 1. Thus, dj0 > r + 1, where djo is the generic dimension of the maximal Segre set of M at P0 9 On the other hand, if we go back to the unscaled cordinates (z, w), we note from the construction of the Segre sets that each N j is contained in the complex manifold X = {(z, w): wr+l . . . . . w m = 0}. Thus djo < r + 1, so that we obtain the desired equality djo = r + 1. This proves part (a) of the theorem. It follows from (4.3.1) that the C R vector fields of M are all tangent to M f q X = {(z, w) e M : wj = O, j = r + 1, ..., m}. Thus, the local C R orbit of P0 is contained in M N X. Also, since there are r H6rmander numbers, the CR orbit of P0 has dimension 2 + r. Since the dimension of M 0 X is 2 + r as well, it follows that the local C R orbit of p0 is M A X. This proves part (b) of the theorem. To prove part (c) of the theorem we note that if M is real algebraic then each Segre set N j is contained in a unique irreducible complex algebraic variety of dimension dj. Since Njo contains a relatively open subset of X, this relatively open subset of X coincides with a relatively open subset of the unique algebraic variety containing Njo. Hence, X is complex algebraic. Finally, to prove part (d) one starts by defining the map Zo(to) to be the constant map with value p0. Subsequent maps are then constructed recursively from the definition of the Segre sets in normal coordinates. We omit the details of this rather simple construction and refer the reader to [BER1, Assertion 3.3.2]. This completes the proof of Theorem 4.1.21. [] 5. AN APPLICATION TO ItOLOMORPHIC MAPPINGS BETWEEN REAL ALGEBRAIC SUBMANIFOLDS 5.1. A r e f o r m u l a t i o n o f T h e o r e m 1.1. We shall reformulate Theorem 1.1 as a local result for germs of holomorphic mappings taking a real algebraic submanifold M into a real algebraic set A ~, and we shall express the conditions on M using the CR geometric notation introduced in sections 2-3. T h e o r e m 5.1.1 ( [ B E R 1 ] ) . Let M C C N be a real algebraic, connected submanifold with Po E M . Assume that M is holomorphicatly nondegenerate, generic, and that there is a point at which M is minimal. Then, if H : C N --~ C N is a germ at po of a hoIomorphic mapping, with Jac H ~ O, such that H ( M ) C A I, where A ~ C C N is a real algebraic set with dim~A I = dim~M, necessarily H is algebraic. Before we turn to the proof of Theorem 5.1.1, we show that this theorem implies Theorem 1.1. P r o p o s i t i o n 5.1.2.
Theorem 5.1.1 ~
Theorem 1.1
Proof of Proposition 5.1.2. Assume Theorem 5.1.1 has been proved. We show that Theorem 1.1 follows. Since Areg is a real algebraic submanifold, it is CR outside a proper real algebraic subset; we denote the set of points at which Areg is CR by ACR- Let M be a component of ACR such that P0 ~ M. We claim that M is holomorphicalty nondegenerate, generic, and that there is a point at which it is minimal. The algebraicity of the mapping H, and hence Theorem 1.1, follows by applying Theorem 5.1.1 to the mapping H with M defined as above.
65 First, observe that M is generic by (2) of Theorem 1.1, because if it were not then it would be contained in a proper complex algebraic submanifold of C N and hence there would be a holomorphic function vanishing on M; indeed, there would be a polynomial vanishing on it. Second, M is minimal at some point, because if it were not then all CR orbits would be proper real analytic submanifolds of M. At a point Pl E M where the dimension of the corresponding CR orbit is maximal, we may apply the Frobenius theorem and deduce that the CR orbits near that point form a local real analytic foliation of M. Thus, there is at least one real analytic, real-valued, non-constant function f on M which is constant on the CR orbits near Pl (in fact, there are l > 0, where l denotes the codimension of the CR orbit at Pl in M, such functions with linearly independent differentials). Since f is constant on the CR orbits, it is CR and, since f is also real analytic, it extends as a holomorphic function in a neighborhood of pl in C g. This violates (2) of Theorem 1.1, since f by construction is real-valued on M. Finally, M is holomorphically nondegenerate, because if it were not then there would be a point pl E M and a germ at pl of a holomorphic vector field tangent to M. It follows from Assertion 3.1.5 (by solving (3.1.6) using Cramer's rule) that then there is a holomorphic algebraic vector field X (i.e with holomorphic algebraic coefficients near Pl) tangent to M near Pl. By following this algebraic vector field into C 2N along the complexification of A,,g, we deduce that this would imply that there is a holomorphic algebraic vector field tangent to Areg at every point outside a proper real algebraic subset of Areg; we refer the reader to the proof of [BER1, Proposition 1.4.1] for the details of this argument. This contradicts (1) of Theorem 1.1. As we mentioned above, this completes the proof of Proposition 5.1.2. [] R e m a r k 5.1.3. Inspecting the proof of Proposition 5.1.2 above, one observes that the term "holomorphic" in the hypotheses (1) and (2) of Theorem 1.1 could be replaced by "holomorphic algebraic" if it were not for the fact that the Frobenius theorem (used to prove minimality of M at one point) only asserts the existence of an analytic function f (see above) and not an algebraic one. However, in [BER1] we prove an "algebraic version" of the Frobenius theorem for the CR vector fields of a real algebraic CR submanifold (by using the Segre sets); we mean algebraic version here in the same sense that Corollary 4.1.31 can be viewed as an algebraic version of the Nagano theorem. By using this result instead of the Frobenius theorem above, we can indeed replace "holomorphic" in the hypotheses (1) and (2) by "holomorphic algebraic". 5.2. P r o o f o f T h e o r e m 5.1.1. Let U C C N be a neighborhood of the point P0 to which the germ H can be extended holomorphically. We claim that we can find a point Pl E M N U such that (a) M is minimal at Pl; (b) M is ~-nondegenerate at pl, where ~ = ~(M) denotes the Levi number of the holomorphically nondegenerate submanifold M as defined following Proposition 3.2.2; (c) H is a biholomorphism at Pl.
66 To see this, recall that a real analytic CR submanifold M is minimal at Pl if and only if M is of finite type at pl. Since M is of finite type at some point, it follows from the definition of finite type that M is of finite type outside a proper real analytic subset. Also, M is f(M)-nondegenerate outside a proper real analytic subset, by Proposition 3.2.2 (iii). Finally, since Jac H 7~ 0 and M is generic (i.e. not contained in a proper complex analytic variety), Jae HIM 5~ 0 outside a proper real analytic subset. The claim above follows easily from these facts. Since H is a biholomorphism at pl, H(M) C A t, and dim~A I = d i m e M , it follows that the real algebraic set A t is a real algebraic manifold M ' at p~ = H(pl) and M t is ~-nondegenerate at p~. Thus by Proposition 3.3.1, H satisfies a reflection identity of the form (3.3.2), where the functions ~ are holomorphie algebraic near the appropriate point. For any j = 1, ...,j0 and tj E C.di near 0, we can substitute (Z, ~) = (Zj(tj), 2j-l(Sj-l(tj))), where Zj(tj) and sj-l(tj) are as defined in Theorem 4.1.21 (d), in the identity (3.3.2). We obtain
(5.2.1)
O'@H(zAtj)) = II_17
Ol~l f i
(Zj(tj), Zj_l(Sj_l(tj)), H(Zj_I(sj_I(tj))),... , - ~
_
(Zj-l(Sj-l(tj))),...),
where I~I < g + 171- Thus, if we know that
OI~IH OZC~ (Zj-l(tj-1)) is a holomorphic algebraic function of t j - l , for every multi-index (~, then it follows from (5.2.1) and Lemma 2.1.2 that
ONH is a holomorphic algebraic of tj, for every 7. Since Zo(to) is the constant map, it follows by induction that, in particular, H(Zjo (tjo)) is algebraic. Now, by Theorem 4.1.21, djo = N and since tjo ~-+Zjo (tjo) is an invertible holomorphic algebraic map, we deduce that H(Z) is algebraic. This completes the proof of Theorem 5.1.1. [] 6. OTHER APPLICATIONS AND CONCLUDING REMARKS 6.1. T h e a l g e b r a i c e q u i v a l e n c e p r o b l e m . As we remarked in the introduction, the conditions that M be (1) holomorphically nondegenerate, (2) generic, and (3) minimal somewhere are also essentially necessary for the conclusion of Theorem 5.1.1 to hold. To be more precise, if (1) or (2) is violated, then there is a nonalgebraic biholomorphism of M into itself fixing any point P0 C M. If (3) is violated and M is weighted homogeneous (with respect to P0), then there is also a nonalgebraic biholomorphisms of M into itself fixing P0; see [BER1] for proofs of these statements. However, instead of asking if all biholomorphisms taking M into M t, and P0 to p~, are algebraic one may ask the following: Suppose that there is a biholomorphism
67
H ( Z ) with H ( M ) C M ' and H(po) = P~o, is there (possibly another) one which is algebraic? This is probably true for a larger class of real algebraic submanifolds than those satisfying (1)-(3) above. Indeed, this is true for all real algebraic curves in C whereas it is not true that all conformal mappings taking one such curve into another are algebraic. (Theorem 5.1.1 does not apply, because a real algebraic curve in C is totally real and hence not minimal at any point.) As far the author knows, this problem has not been much studied, except in the cases where one can show that all maps are algebraic. 6.2. U n i q u e n e s s of b i h o l o m o r p h i s m s b e t w e e n real a n a l y t i c , g e n e r i c subm a n i f o l d s . The technique of using reflection identities combined with the Segre sets can also be used to study germs of biholomorphisms between real analytic submanifolds. Indeed, the idea of proof used to prove Theorem 5.1.1 above can also be used to prove the following uniqueness theorem. T h e o r e m 6.2.1 ( [ B E R 2 ] ) . Let M C C N be a connected, real analytic, and generic submanifold of codimension m, and let Pc G M . Assume that M is minimal at some point, and ko-nondegenerate at Pc. Then, if M I is a real analytic submanifold of codimension m and F, G are germs at pc of biholomorphisms mapping M into M I such that
(6.2.2)
OtatF
~
OlatG
(p0) = - f ~ ( p 0 ) ,
vl~[ <_ (m + 1)k0,
necessarily F - G. Thus, if M is holomorphically nondegenerate, generic, and minimal somewhere then, in view of Proposition 3.2.2, uniqueness in the sense of Theorem 6.2.1 holds at all points p E M outside a proper real analytic subset. One may ask: Does uniqueness hold at all points, or perhaps all minimal points, of such a submanifoId? We have some evidence that suggests that this may be the case. The difficulty at points that are not finitely nondegenerate is that one cannot get a reflection identity of the form (3.3.2) for the mapping (although at so-called essentially finite points a more complicated identity can be derived). Another interesting problem, once one has established uniqueness for germs at P0 of biholomorphisms mapping M into M ~ and P0 to p~, is to describe the image of the set of such germs under the map taking a germ to its jet of the appropriate order #, i.e. (6.2.3)
H~-*
(OH ) ~ - ~ (p0)
9
We refer the reader to [BER3] for results in this direction in the hypersurface case. REFERENCES [BER1] M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Algebraicity of holomorphic mappings between real algebraic sets in C N, Acta Math., (to appear). [BER2] , Infinitesimal CR automorphisms of real analytic manifolds in complex space, Comm. Anal. Geom., (to appear).
68
[BER3] _ _
, Parametrization of local biholomorphisms of real analytic hypersurfuces, (submitted for publication) (1996). [BHR] M. S. Baouendi, X. Huang and L. P. Rothschild, Regularity of CR mappings between algebraic hypersurfaces, to appear, Invent. Math. (1996). [BJT] M. S. Baouendi, H. Jaeobowitz and F. Treves, On the analyticity of CR mappings, Ann. Math. 122 (1985), 365-400. [BR1] M. S. Baouendi and Linda Preiss Rothschild, Germs of CR maps between real analytic hypersurfaces, Invent. Math 93 (1988), 481-500. [BR2] ~ , Geometric properties of mappings between hypersurfaces in complex space, J. Diff. Geom. 31 (1990), 473-499. [BR3] , Mappings of real algebraic hypersurfaces, J. Amer. Math. Soe. 8 (1995), 9971015. [BR4] , Holomorphic mappings between algebraic hypersurfaces in complex space, S6minaire "Equations aux deriv6es partielles" 1994-1995, Ecole Polytechnique, Palaiseau, France, 1994. [BG] T. Bloom and I. Graham, On type conditions for generic real submanifolds of C.n , Invent. Math. 40 (1977), 217-243. IBM] S. Bochner and W. T. Martin, Several complex variables, Princeton University Press, Princeton, NJ, 1948. [B] A. Boggess, CR manifolds and the tangential CR complex, CRC Press, Inc., Boca Roaton, Fla., 1991. [CM] S.-S. Chern and J.K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. [CJ] S.-S. Chern and S. Ji, Projective geometry and R;emann's mapping problem, Math. Ann. 302 (1995), 581-600. [D] P . J . Davis, The Schwarz function and Its Applications, The Carus Mathematical Monographs 17, Math. Assoc. Amer., 1974. [DF] K. Diederich and J. E. Fornaess, Proper holomorphic mappings between real-analytic pseudoconvex domains in C n, Math. Ann. 282 (1988), 681-700. [DW] K. Diederich and S. Webster, A reflection principle for degenerate hypersurfaees, Duke Math. J. 47 (1980), 835-843. IF] F. Forstneri~, Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989), 31-62. [H1] X. Huang, On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions, Ann. Inst. Fourier, Grenoble 44 (1994), 433-463. [H2] __, Schwarz reflection principle in complex spaces of dimension 2, Comm. Part. Diff. Eq. (to appear). [L] H. Lewy, On the boundary behavior of holomorphic mappings, Acad. Naz. Linzei 35 (1977), 1-S. [N] T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398-404. [P] H. Poinear$, Les fonctions analytiques de deux variables et la reprgsentation conforme, Rend. Circ. Mat. Palermo, II. Ser. 23 (1907), 185-220. [Seg] B. Segre, Intorno al problem di Poincard della rappresentazione pseudo-conform, Rend. Acc. Lincei 13 (1931), 676-683. [Serf J.-P. Serre, Gdometrie algdbrique et gdometrie analytique, Ann. Inst. Fourier. Grenoble 6 (1955), 1-42. [Sh] H.S. Shapiro, The Schwarz function and Its Generalization to Higher Dimensions, John Wiley & Sons, New York, NY, 1992. [Stl] N. Stanton, Infinitesimal CR automorphisms of rigid hypersurfaces, Amer. Math. J 117 (1995), 141-167. [St2] _ _ , Infinitesimal CR automorphisms of real hypersurfaces, Amer. J. Math. 118 (1996), 209-233. [SS] R. Sharipov and A. Sukhov, On CR-mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions, Trans. Amer. Math. Soc. 348 (1996), 767-780.
69 [Ta] [Tul] [Tu2]
[TH]
[Wl]
N. Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan 14 (1962), 397-429. A . E . Tumanov, Extending CR funciions on manifolds of finite type to a wedge, Mat. Sbornik 136 (1988), 128-139. , Finite-dimensionality of the group of CR automorphisms of a standard CR manifold, and proper holomorphic mappings of Siegel domains, Izvestia Akad. Nauk SSSR, Ser. Mat. 52 (1988); Math. USSR Izvestia 32 (1989), 655-662. A.E. Tumanov and G. M. Henkin, Local characterization of holomorphic aulomorphisms of Siegel domains, Funktsional. Anal. i Prilozhen IT, 49-61; English transl, in Functional Anal. Appl. 17 (1983). S. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43
(1977), 53-6s. [W2] [W3]
, On mapping an n-ball into an (n + 1)-ball in complex space, Pac. J. Math. 81 (1979), 267-272. , Geometric and dynamical aspects of real submanifolds of complex space, to appear in Proceedings of Int. Cong. Math., Zurich (1994,).
DEPARTMENT OF MATHEMATICS, ROYAL INSTITUTE OF TECHNOLOGY, 100 44 STOCKHOLM, SWEDEN E-mail address: [email protected]
REAL INTEGRAL
GEOMETRY
AND
COMPLEX
ANALYSIS
SIMON GINDIKIN
In these three lectures I will talk about three variations of the Radon transform. We start with the projective Radon transform ( Lecture 1). The projective view gives the possibility to see that the affine Radon transform and the MinkovskiFunk transform on the sphera are projectively equivalent. We increase also our possibilities to write the Radon inversion formulas, transforming them in a projective invarimat form. In Lecture 2 we consider the Radon-John transform - the generMization of the Radon transform when we integrate on planes of the codimension more than 1. We gain here an experience of how to work with overdeterminant problems of integral geometry( the dimension of the manifold of planes can be much higher than the dimension of the space). Finally in Lecture 3 we extend the Radon transform up to the conformal invariant object: the reconstruction of fimctions on the sphera through their integrals on plane sections. This problem is again overdeterminate. In the scope of this problem many problems of the integral geometry are equivalent including the projective Radon transform and both hyperbolic Radon transforms - geodesic and horospherical. In the focus of our exposition is the operator ~ of Gelfand -Graev-Shapiro [GGS]. I believe it is the most remarkable construction in the integral geometry. It gives the way to work with overdeterminant problems of integral geometry. In the practice it is the most universal method to obtain explicit formulas of the integral geometry. It establishes the equivalency of problems of absolutely different nature and explains why explicit inversion formulas for different problems and obtained with often absolutely different methods look similar. The new aspect here is that we consider the operator t~ not only for local problems of the integral geometry but also for nonlocal problems. The way to do it from my point of view goes through O-cohomology and hyperfunctions. It explains the title of these lectures. You will see in Lecture 2 one example of how the integral geometry feeds back the multidimensional complex analysis. I mean the description of the Fourier image of functions (or distributions) with support in nonconvex cones on the language of O-cohomology in noneonvex tubes mad the corresponding generalization of the Laplace transform. I felt a shortage of time and space to expose these fundamental, I believe, ideas. It is why I often give only sketches of proofs and constructions and do not care about computations of coefficients. I want to emphasize that it was an enormous pleasure to talk about integral geometry at Venice. I would like to thank the organizers for this fantastic opportunity.
71 LECTURE 1. THE PROJECTIVE RADON TRANSFORM
1. B a s i c n o t a t i o n s a n d definitions. Let x = (x0, x l , . . . , xn), x # (0) be homogeneous coordinates in the real projective space p n = R p n . It means that we have a projection
(1)
R n + , \ { o } _~ pn,
~ ~ x,
~ c R\o.
Let { = ({0,... ,{n) be homogeneous coordinates in the dual projective space (P~)' = P~, so that
(~,x)
= ~0z0 + ' " + ~ , , x n
=0
give the hyperplanes at p n . Let (2)
~,(x) = r , (-1)% A dx, = det(x, dx,... ,dx) = [ ~ , d x , . . . ,dx] O<j
i#j
be the Leray form. It plays the role of a volume's element in projective computations. We use here the notation d e t ( a 0 , a l , . . . , a n ) for the determinant of the matrix (a0, al ... an) where some of the columns can be the columns of 1-forms and we use the wedge product to compute the determinant. We will write that f E O(m) if f C C ~ ( R n - I \ 0 ) and
f(Ax)
= ~mf(x),
~ e ~\0,
Definition.
We will call
(a)
f(~)= / f(x)5((~,x))w(x)= f pn
x E I~n+l\{0}.
f(x)(d(~,x)JoJ(x))
(~,x)=O
the Radon transform [GGG]. We have ](~) E O ( - 1 ) . In the first integral the integrand-form on R n+l can be pushed down on p n because its degree of homogeneity is zero and it is orthogonal to the fibers of (1). In the second integral the interior product (d(~, z)J0a(x)) is any such form ~ that d((~, x)) A ~ ~(x) =
Such form ~o is not unique, but its restriction on the hyperplane (~, x) = 0 is defined uniquely. We can take
d<~, x>Jw(x)
= det(x,
u, dx,..., dx)
(~,~>
where u is any such vector, that ((, u) # 0 (these forms are different for different u but coincide on the hyperplane).
Remark. The transition out of the first integral to the second in (3) is the simplest example of taking a real residue. We need to explain what means the substitution of the function (~, a) in the distribution 6(t) of one variable. We can do it the naive
72 way: take t = {~.x) as one of the variables, then extend it up a certain way to a complete system of variables, and apply 5(t) on the variable t. It is remarkable that the resulting form on the hyperplane will be independent of such a choice of variables. It is connected with the fact that &function has only a singularity of the first order, and we will see in the future that the situation for singularities of higher order is more complicated. Technically we can integrate in (3) on different sections of (1). Different sections give different interpretations of the projective Radon transform of which we will discuss two. 2 . T h e afflne R a d o n t r a n s f o r m . Let us take the section x0 = 1. Then we have the affine Radon transform of the function V(Y) = f ( 1 , y ) ,
y 9 IR"
Namely if ~ = ( - p , r~) then (4) IR-
where dy = dyl A . . . A dyn = w(1,y),
y)Jd ) -
(-1)~+ 1 7]j
d j, itj
if r5 r 0. Apparently the condition f E O ( - n ) gives ~o a sufficient decreasing condition (0(ly{-'~)) for the convergence of (4). Let us remind you that there is a fundamental connection between the affine Radon transform and and the Fourier transform ] :
(5)
](p,) = & _ j ( , ; ; ) .
where ~-p--0 is the one-dimensional Fourier transform. 3. T h e F u n k - M i n k o v s k l t r a n s f o r m . We obtain such a transform, if we integrate (3) on the sphera x~ + z~ + ... + x .2 = 1 . The restriction of f on the sphera will be an even function f ( x ) on the sphera and we integrate it along great spheras (sections of the sphera by hyperplanes (~, x} = 0 passing through 0). The integral on the sphera S" gives f(~) with the coefficient 2, because the sphera intersects fibers of (1) at 2 points. So we see that the affine Radon transform and the ~mk-Minkovski transform are projectively equivalent. To see the equivalence, it is essential to work with sections of line bundles (w E (.9(-n)), rather then with functions on N n and S n (in the language of functions we need to add a Jacobian). We can automatically transfer all formulas for one of these transforms to another one and do not need to develop independent theories for them. It is interesting to
73
remark that Minkovski and Funk considered their transform earlier than Radon did his. Radon knew about their results, noticed an analogy at his and Funk's inversion formulas (both used Abel's inversion formula), but missed the equivalence, between the two transforms. It is more surprising, that this tradition is alive today, and mathematicians continue to develop independent theories for these transforms. Of course sometimes the work with a smaller invariance group has advantages and the good example is the possibility to invert the Minkovski-Funk transform using the spherical functions (following Minkovski) . I believe that the projective nature of the Radon transform is one of the most important advantages it has, if compared to the almost equivalent Fourier transform. From the other side, the connection of the Radon transform with the projective structure gives the first example of the careful investigation of group invariance and geometrical structures in integral geometry. 4 . T h e affine i n v e r s i o n f o r m u l a for t h e R a d o n t r a n s f o r m . The simplest way to obtain the affine inversion formula is to combine the Fourier inversion formula OO
f(P7) e x p ( - i p ( 7 , y))pn-1 dp A aJ(7 )
f(Y) -- (n - 1)!(2~r) n 0 S~-t
with (5). Using the identity 7(p~ -1) = (~ - i)!i~ (p- io)-" = (~ - i)!i" p -~ -~(-i)" ~(~-~)(p) we obtain OO
(6)
f(Y) =
~
/(7,P)(P - (7, Y) - iO)-ndP A w(T). --00
Sn--1
It is essential that the integrand can be pushed down on the sphera S n-1 rather then on p n - 1 (more precisely, we can integrate it on S n-1 or any other surface F, which intersects almost any ray out of 0 once). Of course we can leave only the even part in the integrand. So we need to take only the even part of (p - i0) - n which coincides with the first term for even n, and the second term for odd n. As the result we have the local inversion formula for odd n and nonlocal for even n. In both cases we can integrate on p n - 1 , but at (6), which is true for all n, we integrate on the sphera. The reason is that ( p - i 0 ) - n is only positive homogeneous. We will interpret the formula (6) the following way. Let us consider the modified Radon transform
(7)
?(7,~0) = c
i](7,p). ---r-~
p t t0
ap,
I m ~0 > 0,
~ = ((0,7).
--oO
It will be holomorphic of (0 on the upper half-plane, C+ = {z : Im z > 0} and it will not be even. Let
(s)
~](zl~, ~7) = c?~: -1)((7, ~), 77)~(7)
74 where
f~(,,-1)_ Of~('*-:) ~0
(O~0p-1
We have the (n - 1)-form on sphera S n-1 which also depends holomorphically on complex parameters z out of the half-spaces C~ = = {z e C ~ : Im(~,z} > 0}. In a general situation, we can not integrate this form on S n-1 because the halfspaces Cg have no joint points for 77 E S '~-1. If tr has boundary values of some sort on R ~, then we can integrate n ] and we will have
(9)
f(x) = c /
n](xl~,d~),
x e ~n
S,-1 For f E O(-n) boundary values exist at least in the $1-sense. Let ](77; p) = 0 if + r / ~ cl(V) and ](r]; p), ~ e If, admits a holomorphic extension on the lower half-plane C_ on p , where V is a sharp (does not contained lines) convex open cone (it is equivalent to the condition supp ] C cl(Y)). Then ](r/; p) for -r] E V is holomorphic at C+ and
?(r/;p) = ](r/;p),
, e v,
? ( , ; p ) - o,
, ~ el(v)
For such f we can extend (9) into the complex domain:
(9')
f(z) = / nf(z[7], &l)
has sense for z E T* = 11~n + iV* where
V* = {y E An: (y,t) > 0 for allt C cl(V)\(O)) is the dual cone to V and we obtain a holomorphic function at the tube T*. The integral on the sphera at (9 ~) reduces to the integral on V N S ('~-1). Conversely holomorphic functions at the tube T* for different functional spaces of Hardy's type-H2(T*), $(T*), ,.q'(T*) etc. (Laplace-dual spaces to L2(V), $(Y), $'(Y) etc.) have the holomorphic (on p) Radon transform satisfying to the above conditions. It is possible to interpret the representation (9 I) as Radon's analog of the Fourier-Laplace transform. 5. R a d o n ' s r e p r e s e n t a t i o n o f h y p e r f u n c t i o n s . In the general situation (9) gives a realization of f as a hyperfunction - cohomology class from H ( n - 1 ) ( C n \ R n, (9). Let us give a realization of this class in the Cech language. If we have a covering of Cn\llU ~ by (n + 1) half-spaces
C~={z;Im(r/j,x}},
j = 1,...,n+1,
75 and Tj* are the intersections Nk#j C~; the intersection of all C ] is empty. Then (n - 1)-cocycles for this cohomology are the collections of holomorphic functions
fj e O(T;)
{fl,. '' ,fn+l}, without any conditions or identifications. elements of H(n-1)(C~\nt ~, O). If
So such collections are precisely the
Tj* = N" +iVj* then it is simple to see that the dual cones Vj are mutually disjoint and the union of their closures gives whole N n. Let
?j Following up on our remark, it will be the Radon transform of a holomorphic function fj at the tube Tj. If we put functions f in the correspondence with the collections { f l , . . . , f~+l }, then we obtain the realization of functions f as the ((~ech) cohomology classes { f l , . . . , f n + l } e H('*-l)(Cn\Nn,O),fj E O(T;). If fj(x), x E R n, are boundary values of fj then
f(x) = f l ( x ) + . . . +
fn+l(X),
x e ~n
We used in this construction the existence of finite coverings of C n \ R " by the Stein manifolds C~. The existence of a finite covering by Stein manifolds for a complex manifold is a very rare phenomena. Let us remark also that the finite covering in our example is noninvariant relative to linear authomorphisms. The Radon transform gives a hint that at cases of infinite coverings it is natural to work with a continuous version of Cech cohomology [G1],[G2] (cf. other examples at Lecture 2). Namely we have the covering of C n \ N '~ by half-spaces C~ = {z 6 C", Im 0},
~ 6 S "-1
So S n-1 is the parametric manifold. Let us consider differential (n - 1)-forms
dr/) which holomorphically depend on z E C~ as of parameters. So parametrical domains differ for different r/E S n-1. Let us consider the complex of such forms with parameters. In our case forms w have maximal degree and are automatically closed so we need only to factorize their space of all forms on the subspace of exact forms. It is possible to prove that this quotient is isomorphic to H ( n - 1 ) ( C n \ R ~, (9). It is the consequence of a very general result about the continuous Cech cohomology (it is important only that the submanifold Sz of parameters r/corresponding to the Stein domains C~ containing a fixed point z is contractible). The Radon transform gives a very special representation of f E O ( - n ) as a form aJ(z]r/, dr/). Namely, the form ~;f(z]rh drl) as a function of z depends only on (r/, z}. on S n-1
76 It is possible to interpret the Radon transform as a tool to obtain such special representations of hyperfunctions. If f E S, then such a representation ,; does exist, and is unique if it is put on w(z] .) as a function of z the natural S-conditions of smoothness and decreasing and moreover there are no exact forms with such conditions. The same is the situation with f E L 2 either f E C' or other space of decreasing distributions. We observe another situation for f E S ' : nontrivial exact forms w with S~-restrictions exist and they correspond to nonessential distributions in the sense of Gelfand - Graev [GGV]. It is usually the case that the description of the Radon image of spaces of increasing distributions must include a factorization and a cohomological language is very natural for this description. It would be interesting to investigate the general problem of "Radon" representations of cohomology-hyperfunctions. There is a very simple explicit (chain) morphism out of continuous Cech cohomology to the corresponding Dolbeault cohomology. Namely let y = 7(z) be any function (section) o n c n \ ~ n such that
Im(~(z),z) > 0. Then
(~o)
~ ~(zl~(z), d~(z))] ( ~
induces a morphism out of the continuous cohomology to the Dolbeault cohomology. At (10) we restrict w on a section ~ -- 7(z) and then take (0, n - 1)-part of the resulting form on c n \ R n. 6. P r o j e c t i v e i n v e r s i o n f o r m u l a . We want to give a projective invaxiant version of the inversion formula. The first step is to replace our affine inversion formula so that the formula will depend on any plane on infinity ((,z) = 0
z EPn
and our affine inversion formula (9) will correspond to ~ = (1, 0 , . . . , 0). It is simple to rewrite formulas. Let
(11)
T(z] + A~]~) ----c
(12)
~](zl~,d~lr
/ ](7p - -+~Pr dp,
q, ~ E P n , I m A > 0.
1 ( d ) n-1 = (r
f~(~ + Ar162162
,d,],
Here z is a point of the affine semispace T,,r
(13)
(r/, z)
T,,r = {z E CPn; Im ~
< 0}.
(~,z) A = - - -(r z)'
77 If F(() is any cycle intersecting any ray out of ( once (e.g. sphera Srn--1 with the center ~), then 2r~,r and ~ ] depend only on a ray through r / a n d the family of is a covering of (14)
Cn(~)\R"(~),
C " ( ( ) = C P n \ { ( ~ , x ) = O}
R"(()=Cn(~)
Tn,r
N RP"
So ~ ] gives a cohomology class (on the language of continuous Cech cohomology) at C n ( ( ) \ Rn((). If x ] has boundary values on z = x E lI~n(() then we have an inversion formula f
(15)
f(~) = c [
~](xl,,d,),
x ~ R"(().
r(~) We obtain the affme formula (9) for ( = ( 1 , 0 , . . . , 0). Of course this construction is already projectively invariant, but it is much more interesting to find a formula where the affmization ~ is variable (depends of r/). All forms in formulas (12) represent the same functional for different ~, but they do not coincide for different (: they differ on exact forms. The idea is to construct a closed form ~ ] so that it includes differentials d( and for fixed ~ coincides with (12). It is possible to compute such an extension using only the closure of the form, but we will do it using another way. So our goal is to give an invariant sense to the functional
(16)
f/(~)(<~,~)
-/o)-"~(~).
Np-
Let us consider a = ](~)(~, z ) - " ~ ( ~ ) ,
z e CP".
If (~, x) 56 0, then this form will be closed (and can be pushed down on S n or a homological cycle). Let us substitute ~ = 7? + p~. We obtain the closed form
a - ] ( ' + ;~) ~ ( ' + ; ; )
<~, z)-
~-
(p - ,~)- '
<'' z) (r
We suppose that z E To,r so Im A > 0. Let us integrate this form on - e ~ < p < cx~. The result will be a closed form of (rh (). It is a corollary of a general fact that if we integrate a closed form of a degree s on some cycles of a dimension r < s, then the result (direct image) will be the closed form of the degree s - r on the manifold of the cycles. We are going to integrate along afflne lines, but the integration can be extended to the projective lines (cycles). Let us compute it through f ( r / + A(I(). We need to know
dpJw(rl + P() =
Z O<_k
akpk[rl'('d([k]'&l[n-l-k]]'
n - 1) ak---=n
k
78 and
pk = ~_, b,.k~k-,.(p_ A)r., b~k = (~). O
Here d( [k] means that we repeat the column d( in the matrix k times and similarly for drl['q. Finally (17)
tc/(Zlrh~'d~'d~) =
E
Pk ( 0 ) T ( ~ +
As
In-l-k]]
O
=
~k
kg~/
,
c ~ k = (n - .~ - 1)! ak bm~,
o<m
~ = <~' z>
(C, z)"
This form is closed and represents a cohomology class on CP"\NP n (half-spaces T~,i form a covering). If r = const then we have only the term with dr/ which coincides with (15). If we take boundary values on N:P", we obtain the inversion formula with variable affinization:
(18)
f(x) = ef~f(xl~,C,d~,dC),
(C,=) # o.
F
Here I" is any cycle at 07, () over the hyperplane (r/, x) = 0 with the restriction
(r # o. Remarks. I. For odd n the inversion formula is local. It is a good exercise to write the local projective invariant inversion formula for this case. 2.It is natural to define such a family of transforms, that the Radon transform as well its inversion would be a part of this family. Let f E O(-k), ] < k < n and (19)
TCkf(() = ~S'~ f(x)( (~, x) -- iO)-n+k-l w(~).
The Radon operator corresponds k = n, and its inversion to k = 1. We have (20)
~"~k
0
T~n_k+ 1 = const
Id,
where the general proof does not differ from the case k = n. 3. There is an important analogy between the Radon inversion formula and the Cauchy-Fantappie integral formula for holomorphic functions of several variables. There is a formal way of transforming some facts about one of these formulas to the similar facts about another one. Roughly speaking the Cauchy kernel (one dimensional) and its degrees at the Cauchy-Fantappie formula correspond to the distributions (p - i0) -~ at the Radon inversion formula [G3]. In this way our computations of the projective Radon inversion formula are parallel to the results of [GH] on the projective Cauchy-Fantappie formula. 4. Another curious construction is the class T of domains which are intersections of Tv, r This class is a projective "envelope" of the class of tube domains (which are affine objects). For such domains x f gives a representation of holomorphic functions
79 (cf.(9')). We can interpret this representation as the R a d o n - L a p l a c e transform. Of course the Fourier-Laplace transform does not exit outside the class of t u b e domains. I believe t h a t it makes sense to systematically develop the analysis in these "projective t u b e domains". Let us summarize our conclusions. From the point of view of the theory of distributions, the construction of an inversion formula is the p r o b l e m on a realization of the distribution c((~, x > - i0) - n on the test space O ( - 1 ) o n P~. This distribution is R a d o n - d u a l to 6(y - x). Informally under realizations we u n d e r s t a n d such representations of distributions as measures, densities, differential forms etc., depending of test functions, t h a t their integration gives evaluations of distributions on the test functions. Of course realizations are non unique and we need to choose apriori some tools for realizations. So formulas and realizations are basically synonyms in our context. We could have o b t a i n e d inversion formulas by using the densities, b u t we preferred to consider differential forms which are holomorfically dependent of p a r a m eters out of variable domains. Here we have the freedom on exact forms which is s t a n d a r d for cohomology. Let us emphasize t h a t this language uses e x t r a structures on the projective space - affinizations. This phenomena is very f u n d a m e n t a l for multidimensional analysis and similar to s u p p l e m a t a r y structures for computations of residues for singularities of higher order for several complex variables [GH].
80 LECTURE 2, THE RADON-JOHN TRANSFORM 1 . B a s i c n o t a t i o n s a n d d e f i n i t i o n s . We will call the o p e r a t o r of integration on q-dimensional planes in the projective space p n the Radon-John transform of dimension q. For q = n - 1 we have the R a d o n transform; F . J o h n considered the integration on lines in R 3. So let f E O ( - n + r - 1) on p n and ~ = < ~ O ) , . . . , ~ ( r ) > , r = n - q, be a system of independent vectors. We will write (~, x) = 0 instead of (((a), x> . . . . . (~(~), x) = O. T h e n
](() = ]((1~,..., (,)) = p,,f f(x) e(((1), x>... a(((~Lx)) ~(~) (1) <~,.~>=o
J'
We o b t a i n the affine version of this transform if x = (1, y), ~(J) = (-p(J), rl(J)), ~ ( y ) = f ( 1 , y) a n d again (r/, y) = p means that (r/(1), y)) = p f l ) , . . . , (r/(r), y) = p(~). So
(i') There is a dual way to define p-planes as the set of
{x = to uO + t l ul + . . . + tqu q ~- tu, t E Pq} where { u ( ~
(2)
u (q) } is a system of (q + 1) independent vectors. Then
~(u) =/v(tu)~q(t),
~e
O(-q
-
1),
q =
,~ -
r.
pq Of course there is the s t a n d a r d way to transfer out of one p a r a m e t e r i z a t i o n to a n o t h e r one, but it is useful to have formulas for b o t h of them. 2 . I n v e r s i o n f o r m u l a s a n d t h e o p e r a t o r x. For simplicity we will work now with the a ~ n e R a d o n - J o h n transform (the projective generalization is a g o o d exercise). Let ~ E S(N'~). We will start with the connection between the Fourier a n d the R a d o n - J o h n transforms. We have
(3)
@(pr/) = .)C'p_..p(@(r/;p)),
pr/ = plr/(1) -{--999 + PrTI(r),
where the r-dimensional Fourier transform 9v corresponds to the duality form
(P, P) = PiP1 + "'" + P~P~. The proof is the same as for r = 1 and we will not reproduce it here.
81
Let us try to inverse the R a d o n - J o h n transform using the same idea as for r = 1. We take the Fourier inversion formula ~(y) = c L , ~(C)exp(-i 1 we have "too many" planes. We will use the same trick as in the end of Lecture 1: we will pull back the integrand on the fibering (p, r/) --+ ~, integrate the result along R~ and obtain a closed q- form on the Schubert manifold of the r-frames 7/. More precisely we rewrite d( as d~ = ~t. d e t ( d < , . . . , de) = n)-~.[d~',-99 d(], substitute ~ = p r / a n d separate the part of the form containing dp = dp1 9 9 9dp~ :
cdp A [r/(l ,,.
,r/(r)
(pdr/)[q]]~(r/;
p)Ip_-<.,=>.
where the column p d r / = pldr/(1)+ 99.+prdu (O repeats q times. After the integration on R~ we obtain using (3) the q-form (4)
~(y;
'
r/, dr/) = c[r/(1),
'"
., r/(r), (__0 dr/~tq]l '0p ' J ~(r/;P)[p=<~,~>"
This determinant is the form on 77 whose coefficients are some differential operators on p evaluated at p = < r/, y >. This form on the Stiefel manifold of cycles (r-planes with the frames 77) will be closed as the result of the integration of the closed form (of maximal degree) along cycles (~ E S). We will note through ~ ( p ; r/, dr/) the form in (40 before the substitution p = (77, y). We now want to reconstruct ~ integrating of this form on appropriate cycles 7. The situation with the choice of the cycles is not simple. It is simple to prove that always
(5)
f
r/, dr/) = c(7) v(v)
where c(7 ) depends only on the cycle 7, but this coefficient can be equM to 0 and then we can not reconstruct T. It turns out that for odd q the coefficient c(7 ) = 0 for all cycles 7 and we have no inversion formulas. However if q is even, then the coefficient is not equal to 0 for generic cycles and we have the big selection of the inversion formulas (which are parameterized by the cycles 7). The important point is how to compute the coefficient using the geometry of cycles. We can remark that the Stiefel manifold of frames (r/) is the covering of the Grassmanian G r ( r , n) (r-subspaces at R n) and the form ~ can be pushed down on the G r ( r , n). W e need to compute the integrals on basic cycles of the homology group of the Grassmanian: Schubert's cells. They will be 0 for all cells, but the
82 Euler cell, which consists of the r-supspaces at the fixed r + 1-subspace and c(7 ) is proportional to the intersection index of 3' with the Euler class. There is the naive way to do the same things and it works perfectly in examples. The image of 3, in II{n (with the orientation) is proportional to all space and the coefficient is just c(3~) up to a constant factor. In other words, we need for q-plane {7} generated by a generic point 7 E 7 to compute the number of hyperplanes containing {7} with regard to the orientation (we consider here the planes passing through a fixed y E R"). For odd q every hyperplane will appear the same number of times with opposite orientations and as the result the coefficients are always equal to zero. Let us discuss some simple examples. If r = 1, we obtain the usual form
~rt--1
c
p)Ip_-<,,y> [7, dT,...,
in the inversion Radon formula for odd n (for even n this integral on the sphera is zero). Let us now suppose that the last r - 1 vectors 7(2),... ,7 (~) are fixed , let say 7 ( 1 ) = 6 (q+j), j > l ; 6~k ) = l , ~}k)=O, i#k. We have the cycle of all q-planes in the fixed (q + 1)-plane j > 1}
{y : yq+j -= p(J),
(the Euler cycle) and the restriction of a~ on this cycle will coincide with the form in the Radon inversion formula for this (q + 1)-plane: Oq c--@(70),~ 0(p(1))q
(q+2) .,~(n);p(1),yq+2,...,yn)lpO)=(r ''"
[ (1) d (1) , (1) ] x [r/[q+ll , r/[q+ll,...,a7[q+l] ]
7Is] = ( 7 1 , . . . , 7 s , 0 , . . .
) ,0).
The next example is a little bit more complicated: n = nl + n2, q = ql + q2, qj < nj,
rj = nj - qj;
q(j) = ~(q2+J),
rl(J) = ~(ql+j),
1 < j _< nl;
nl+l<j.
The inversion formula for this cycle reduces to the pair of the Radon inversion formulas in the pair of transversal planes of dimensions ql + 1 and q2 + 1. This cycle 3' for n = 4, q = 2, nl = n2 = 2, ql = q2 = 1 is not homological to 0, ~93 is not exact (q = 2 is even), but c(7) is 0. Remarks. 1. The closure of the form t@ is the consequence (and moreover equivalent) to the system of differential equations
(6)
o 2]
(i)
(k)
0(j 0(,
(i)
(k) --
0( t 0~
o
83 It is simple to obtain (6) by the direct differentiation of (1) (but it is not simple to derive the closure of g~3 out of (6)). We will call (6) by the equations of F.John (who defined them in the case n = 3, q = 1).It is possible to prove that the space of the sections F ( ~ I , . . . , ~ ) 6 O ( - 1 , . . . , - 1 ) satisfying to (6), is exactly the image of O ( - q - 1)) on P " for the Radon-John transform (we consider C r162sections). For r = 1 the system (6) is trivial. It is the Paley-Wiener theorem for the projective R a d o n - J o h n transform. The affine version of this theorem image of S(•"). If q > 1 and we put the precise definition), then the John If q = 1 then (6) is empty, but there [GGV]): momentums
is more complicated. Let us investigate the the natural S-conditions on r (we drop system (6) gives the description of the image. are the Cavalieri conditions (Gelfand-Graev,
xk~(~) = j ~(,7;p)pk@ --00
are polynomials on 7? of degree k. These conditions are equivalent to the smoothness ~(~) at 0. For q > 1 it is also possible to formulate the analog of the Cavalieri conditions, however these integral conditions are the consequence of the differential conditions (6). This remarkable theorem was proved by Gelfand-Graev-Shapiro
[CGS]. 2. In the integral geometry we usually consider the complexes r (n-dimensional submanifolds) of q-planes and we want to reconstruct ~o only through c~li-. If 7 = Fv is the submanifold of planes from P passing through y, then for generic y EII{n we can expect that dim 7 = q and if it is the cycle we can use (5) to reconstruct ~(y). The most serious obstruction is that as a rule we can not compute the result of application of the differential operator ~ to qo for r] 6 7 only through the restriction of ~ on F. If such computations are possible for all y the complex r is called admissible. It is a very degenerate condition. In a sense P must be characteristic
for (6). The simplest class of admissible complexes is the class of translation-invariant complexes. So we can start of any q-cycle 7 of q-planes through a point y and take the complex P of all q-planes parallel to the planes out of 7. These complexes are admissible and if c(7) # 0, then we can reconstruct ~ through ~31r.
The admissible complexes of lines are classified, however admissible complexes for q > 1 are not classified and moreover it is not easy to construct explicit examples of such complexes. Let us mention only one class of examples [G4]. Let A be a q-cycle in the Grassmanian Gr(q, n + 1) (the manifold of (q - 1)-planes at p n ) and F(A) be the complex of q-planes which contain at least one of planes of A. Then the complex F(A) is admissible. The special case of such complexes for q = 1 is well known: complexes of lines intersecting a curve. Admissible complexes are very rare and are a degenerate class of complexes, but they appear in some important applications: representations theory, nonlinear differential equations.
84 In dual parameters operator n has the following form:
(7) kT(x;ul,...,uq,du('),...,du
(q))= A \ou(O)
o +...+,:,
5> 0
=
~_, I=(il,...,iq)
(o) r
o ~ ( ~ ) (o) d~!~ . ~, ,, h . . . A a u q h,(o)=x,
., ( o ) - - ' - ~ O'Uil
n i ..... /
x E
~tn.
' " " UUiq
We differentiate here on the coordinates of u ~ and take differentials of the corresponding coordinates of other vectors u i in the special order. As the result this form of dr/(1),..., dr~(q) which depends on parameters x E R n, is a combination of differentia/monomials of the very special form. Such combinations can be characterized by the following condition: they are zero on any submanifold of planes containing a fixed point. In such a form the operator ~ was introduced by GelfandGraev-Shapiro. This representation is projective. It is possible to derive this formula out of (4) but it is simpler to investigate this operator independently. By the direct differentiation of (2) we obtain that
02~ (8)
~
(0~
02~
(k)
ouj ou I
(k)_~ (0
~
r
= 0
uu!
and then again directly we can see that
d(~)
= o
so ~ is closed (we take the differentials only on r/(J), j > 0). We can check the analog of (5) for the base cycles. Usually this version of ~ is more convenient for computations. 3. N o n l o c a l i n v e r s i o n f o r m u l a s a n d O- c o h o m o l o g y in n o n c o n v e x t u b e s . Ir~ Lecture 1 for r = 1 we constructed the universal inversion formula using (p - i0) - n which works in both local and nonlocal cases. Unfortunately this construction does not generalize for r > 1 at the full volume. To understand obstructions in the way we will interpret the operator ~ in the language of densities. Let us compute the evaluation of the form ~ ( p , r/; dr/) on some r-dimensional tangent subspace at the point r/0. It is convenient to identify locally this subspace with the r-plane L passing through r/0 (at r/-coordinates). If dT1,..., dvr are some coordinates on this subspace and ~ = pr/where r/E L, then at r/0:
d~ = ,](p) dp dr The Jaeobiaa J is the polynomial on p which depends on 7/0, L and J ( p ) is the symbol of the differential operator which gives ~ on L:
(9)
0
~ ( y , r/0, dr/)lL = Y(~)~(r/0;
P)]p:<~0,~>"
The trouble is that for the reproduction of the integration on ~ we need to integrate IJ(P)] rather than J ( p ) . When q is even, it can be seen to occur that the
85 J a c o b i a n J(p) conservs the sign along a cycle r/E 7 and then by untegration of ~q3 on 7 we will inverse the R a d o n - J o h n transform. In the general case we can replace the differential form ~3 by a n o t h e r analytic object - the even q-density A~ whose evaluation on q-vectors will use the pseudodifferential o p e r a t o r with the symbol [fl(p)[ instead of the differential o p e r a t o r with the symbol ,](p). The q-density A~3 which in the contrast to the differential form ~3 will not be a multilinear form on dr/. The inversion formulas using the q-densities were considered in [GG]. In Lecture 1 for r = 1 we replaced the integration along lines with the integration along rays where the Jacobians have constant signs and and as the result we worked with the differential forms on the manifolds of rays S (~-1). This trick does not works for r > 1 for m a n y reasons: we can not divide r-planes on non trivial convex cones, a n d also the domains where Jacobians have constant sign d e p e n d on L etc. However there is a special situation when it is possible to realize this plan for r > 1 a n d it brings interesting consequences. Let V be an open cone in R n. It is essential t h a t we do not suppose t h a t V is convex. L a t e r we will p u t some strong restrictions on V. You can be guided by the following 3 examples: 1. T h e qudratic cones V(q+l,r-1)
= { ~ e l ~ n ; ~ 2 + . . ' + ~ q2+ l - ~ q +2~ . . . . .
~n2 > 0},
r + q = n, q > 0.
If q = 0 then V(1, n - 1) has 2 connected convex components: the future and past light cones. 2. V = M+(m) consists of all matrixes of the order m with the positive determinant. 3. V = Sym(~, m - s) consists of all symmetric matrixes of the order m with the signature (8, m - s). For s = m or s -- 0 this cone is convex. Let us consider a space of functions or distributions r with s u p p o r t s in V (e.g. r E L2(V), or S ' ( V ) ) . The problem is to describe the Fourier-dual spaces. If the cone V is convex, then there are known P a l e y - W i e n e r - B o c h n e r theorems of different kinds. T h e y give the descriptions of the corresponding H a r d y t y p e spaces of holomorphic functions in tubes T* = R n + iV* (cf. Lecture 1) a n d the F o u r i e r - L a p l a c e transform establishes the isomorphisms between the two series of the spaces. Roughly speaking H2(T), $'(T) etc. consist of holomorphic functions in the tubes T which have the b o u n d a r y values on R n in the topology of the corresponding spaces L 2, $ ' etc ( a b o u t H2(T) cf. below). We will describe below some analogs of these results for some nonconvex cones [Gb]. For the definiteness we will talk about L2(V) and later we will make some remarks a b o u t S'(V). Let us consider some geometrical constructions for nonconvex cones. If the intersection of V with an r-subspace L has the connected component V(L) which is a convex sharp cone in this plane, then we will call V(L) by the r-slice of V. Of course one subspace L can contain several slices. A c t u a l l y we are interested in r-slices for such a m a x i m a l r t h a t the union of the r-slices coincides with V. Moreover often we do not consider all slices, b u t only a set sufficient for a covering of V. In E x a m p l e 1 we have r-slices - 2 components of sections be the subspace {~2 . . . . . (q+l = 0} or more general V(w) = {~; ~+ =
86
Aw, A > 0}, where ~o E Sq and (+ = (~1,... ,~q+l, 0,..., 0). This family of slices is a l r e a d y sufficient for the covering, but we can extend it by the action of the group SO(q + 1, r - 1) of the a u t h o m o r p h i s m s of the cone. There are no/-slices for l > r. In E x a m p l e 2 the cone V(e) = Sym(m, 0) of symmetric positive m a t r i x e s is the r-slice, r = m(m + 1)/2. We will o b t a i n the slices Y(gl, g2) if transform Y(e) by the a u t h o m o r p h i s m s of V:
~ gl(g2,
gl, g2 E GL(m; N)
In E x a m p l e 3 let us consider the subcone V(e) of the matrixes
(A
BO) '
A E Sym(l, O), B E Sym(O, m - l).
Here A are symmetric positive matrixes of the order 1 and B are symmetric negative matrixes of the order r n - l . It is the r-slice with r = l(l+l)/2 + ( m - l + l ) ( r n - l ) / 2 as well as its translations V(g) by the a u t h o m o r p h i s m s of V
~ g~gT,
g E GL(m; R).
For general cones we will parameterize the r-slices in the following way. Let H~(V) = I I ( Y ) be the manifold of such r-frames r] = {~J} t h a t the positive r - e d r o n
{~ = P~;Pl > O,...,pr > 0 } contains one and only one r-slice V(q). Then to each 7] E I I ( V ) there corresponds only one slice V (but different r / c a n give the same slice V(r/). T h e image of V(rl) at the coordinate space N~ relative to the base r1 will be a cone v(rl) inside the positive r-edron. Let (10)
~(r]; p) = ~-p--p~2(P~)ipev(,)"
This function as the function of p E R ~ will be extendable as a holomorphic function at the t u b e t*(~) = W + i v * ( , ) C C ~, where v(r/)* is the cone dual to v(rl). Let us now (cf.(4)) put
(11)
= c[,(1),.
( )Iql] ?(,;
This form as a function of the p a r a m e t e r s z will be holomorphic in the convex t u b e domain (12)
T*(q) =I~ n +iV*(~),
87 where V*(r}) is the dual cone to the convex (but non open) cone V(r/) C R n. Since the cone V(r/) is contained in the r-subspace {r}} the cone V*(r/) will be invariant relatively to all translations in the direction of the orthogonal subspace {r/}Z(the cone V*(r/) will not be sharp). We have (13)
Y*(,) = {y E R"; (,, y) E v*(,)}
In (11) we obtain objects which we considered in Lecture 1 - the differential forms with holomorphic parameters. It is natural to connect them with q-dimensional O-cohomology of the nonconvex tube (14)
T* =
n + iV*,
where V* = U v * ( , ) . yen
We can consider V* as the analog of the dual cone for nonconvex cones. Let us call cones V*(r/) by the wedges of V*. We obtain the operator to the cohomology in quite a general situation, but without strong conditions on the geometry of the cone V the induced operator on the cohomology can have a big kernel (or sometimes can be trivial) and we do not have the correspondence between functions on R n and cohomology. It is not a surprise that we can not work with arbitrary nonconvex cones. Now it is the time to define a class of "good" noconvex cones.
D e f i n i t i o n . The cone V is called regular with the convexity index r (the concavity index q -- n - r) if there is such a q- dimensional cycle 7 C II(r) that (i) = v. (ii) The slices V(r/), 77 E 7, are mutually disjoint. (iii) For any 3 slices Y(~), Y(r/), V((); ~, r/, r C 7, the slice V(r either is contained in the convex envelope Y(~) + Y(r/) of Y(~) and Y(r/) or does not intersect with it (concavity condition).
Under these conditions we will call the cycle 7 as the concave generating cycle. We will work below only with regular cones V and concave generating cycles 7. In the definition of V* we will replace II on 7. In the regular case r-slices V(r/) are sharp convex cones of the maximal dimension which are component of plane sections of Y. Their dual V*(r}) (wedges) are the maximal convex subcones of V*. The condition (ii) means that the convex envelope of any 2 wedges is the whole space R n. The condition (iii) is the dual of the condition that any wedge can either contain the intersection of 2 other wedges or the convex envelope of it and the intersection is the whole space. It would be interesting to develop the geometry of the regular cones. Several problems at first: To prove that the construction of V* where in (14) we take a generating cycle 9' instead H is independent of the choice of the cycle 7 and coincides with the construction (14). I believe that it follows out of some analytical results on hyperfunctions and wave fronts, but it would be interesting to give a direct geometrical proof. Another question: is the cone V* regular if V is regular? If the answer is positive then is it true, that (V*)* = V? Let us turn to our examples:
88
1. For the cone V(q + 1, r - 1) the family of r-slices V(w) will be the cycle 7,2_ S q satisfying all conditions of the concave generating cycle. The dual cones V*(w) - wedges of V * - are obtained out of the slices by the direct product with the orthogonal q-subspaces. It is remarkable that their union will again give the cone V(q + 1, r - 1). So this cone is sel~dual relative to the defined duality: V = V*. 2. For V = M + ( m ) we parameterize points of the cycle 7 by the elements u of the orthogonal group S O ( m ) and take the slices V ( e , u) The verification of (i) and (ii) is trivial, of (iii) is the exercise in linear algebra. If to use the duality on the matrix space M ( m ) by the bilinear form
(r v) = tr (r y), then V*(e) = {x 9 M ( m ) ; (x + x c) 9 Sym(m, 0)}. So this cone is the direct product of the slice V(e) on the subspace of skewsymmetric matrixes, (q = m ( m - 1)/2). It is a good exercise to prove that this cone will be inside M +. Correspondingly V*(u) = u. V*(e) and we obtain that the cone M + is self-dual. 3. For the cone in Example 3 slices V ( u ) , u 9 S O ( m ) , give the concave generating cycle 7. We have V ( g ) = V(e) if g =
(0" 0),
A'9
K
Therefore the cycle is isomorphic to S O ( m ) / S O ( 1 ) • S O ( m - l) and dim 7 = q = t • (m - l). This cone is also self-dual. We had considered 3 classes of self-dual homogeneous cones; about such cones refer to [FG]. Let us mention that the cone V = R n \ {0} is a self-dual regular cone. The rays are its 1-slices V(r/), r / 9 S "-~, the half-spaces are the dual wedges V*(r/) and the only regular cycle is the sphera S n-1. In this way the results in the end of Lecture 1 are the partial case of the results of this Lecture. For any generating cycle 7 we have
(15)
f ~ ( x , 7, dr/) = c(~)~(x),
~ e L~(V),
x 9 ~tn,
c(~) # 0.
3~
We take here the boundary values of n ~ ( z , r/, dr/) in L2-sense. We need only to remark that by virtue of the condition (ii) of the definition of regular cones the Jacobians f f will have no zeroes on V(r/),7? E 7, and it means as we already remarked, that the integral at (15) gives the integral on l~n at the Fourier inversion formula. Thus we connected with ip(x) the form representing a continuous Cech cohomology class in the tube T* relative to the covering {T*(r/), r/E 7}. For this manifold we can not avoid the use of the infinite Stein coverings (depending on continuous parameters). Let us look on the usual Cech cochains for our covering. Suppose for
89
the simplicity that q = 1. Using the concavity (iii) of the cycle 3' we can define an ordering on the ~,: -~~77 if V ( ( ) C ( V ( ~ ) M V ( r / ) ) Put
j([~
a ~ ( z ; r de),
r
') =
z 9 T*(() fl T*(r/).
,,1
By virtue of (iii) we have a 1-cochain and moreover r
•) = r
() + r
U)
onT*(~) M T*(rl) = T*(~) a T*(q) M T*(~)
It shows that the system of r ~) is extremely overdeterminate and the transition to continuous cohomology is a way to rationalize this information. It is convenient sometimes to use some quasipolyedron approximations. Let X = { t / ( i b . . . , q(N)} be such a cyclic sequence of points in 7, that between points with sequential numbers are no other points of AF. Let us put
VN =
U
V(i),
where V(i) = V(rl(i) ) + V(q(i+l))
I
= { ~ { ( z ) 9 O(Ti%), 1 < i < N - 1}.
Following the remark above, this collection can be uniquely extended up to a Cech 1-cocycle. We have no restrictions on wj so there are no restrictions on parts of 1-cocycles corresponding to the pairwise intesections T(}) = T*(U(0 ) ~ T*(rhi+l). tf we have a set (16) we can define the boundary values of the cohomology class on R '~ taking the boundary values wj(x) and puting (17)
~(X) = ~ I ( X ) + ' ' ' + ~ n _ I ( X ) ,
X C ~n
Here we can take the boundary values in any sense including in hyperfunctions. Let us remind that every holomorphic function in a convex tube T has boundary values on the edge N '~ in hyperfunctions. To justify the definition (17) we must be sure that if a;(x) = 0 then class w is cohomological to zero. If we work with functional space of Hardy's type then we can take the Fourier transform &j(~) which have supports in cl(V(j)). It follows out of (17) that &j(~) has supports in V(q(i)). Then at L2-case we obtain that all ~j - 0 and for 8 ' it simple to construct a 0-cochain for which w is the coboundary. For the hyperfunctions this fact is the simple corollary of the edge of the wedge theorem in the Martineau form.
90 As the result of our constructions we can to define Hardy-type spaces of cohomology in T~: we need only to require that wi belong to the corresponding Hardy spaces in the convex tubes 7'(*). What about T* it is sufficient that the Hardy condition would be satisfied for the restriction w on T~, but it is better to have the direct constructions on the language of continuous cohomology. Let us define the Hardy space of 0-cohomology For the convex tube the Hardy space H2(T*) is selected out of the space of holomorphic functions ~(z) in T* by the L2-norm
H~q)(T*).
(18)
II ~ I1~ = supv6v* II ~(" +
iv) II2
where we take for fixed y usual L2-norm on R". For q > 0 we need to compute the dual norm for
II ~(p~)II,
~ ~
L~(V)
relative to .Tp__.p. We need almost to repeat our computations for (4). Namely we need to apply the Parseval formula on (p, p) keeping in mind the Jacobian if(p) :
(19) II O(p,D ll2= c II ~ 11}4~ f
[
Im
sup
pEV* (rl)
~m
~(p,
7?d~) 9 ~(~; p) dp]
p=conat
For the same reason as for the inversion formulas the form which we integrate on 7 in (19) is closed. Therefore we can in the definition of the norm (19) integrate on any cycle homological to the regular cycle.
91 LECTURE 3. THE CONFORMAL RADON TRANSFORM In Lecture I we extended the affine Radon transform up to the projective R a d o n transform. It corresponds to the projective compactification of the affine space R " up to the projective space N P ' . In the process of the compactification we a d d to R" the line at infinity and extend the group of affine transformations up to the projective group SL(n + 1; R). There is another remarkable compactification of N '~ - the conformed compactification NC n. In this case we a d d only one point at infinity and extend the affine group up to the conformal group SO(n + 1; 1). T h e cop.formal space N C n can be realized as the sphera S n :
(1)
x~ + ' ' '
2 1 =1 -I" Xn+
The e m b e d d i n g of R n in NC n is the inverse m a p to the stereographie projection out of the north pole N = (0, . . . 0, 1) which corresponds to the point at infinity. Sections passing t h r o u g h N and only they correspond to hyperplanes. To see the action of the eonformal group we consider the projective coordinates in which S " corresponds to the quadric Q:
(1')
x n+l 2 =o
.....
and the group SO(n + 1; 1) of projective transformations conserving [](x) is exactly the conformal group. W h e n we transfer from the affine geometry to the conformal one we e x t e n d the family of hyperplanes up to the family containing all h y p e r p l a n e s as well all as spheras. This geometry contains as a subgeometry not only the affine geometry but also the hyperbolic geometry. Let us consider the affinization x,~+l = 1 where we o b t a i n the affine h y p e r b o l o i d Hn:
(2)
x2
_
x 2n = 1
x2 .....
and the hyperbolic geometry is realized on one of its two sheets H_~ (x0 > O) (or we can identify the symmetric points) with the group of hyperbolic motions SO(n; 1). T h e projection out of ( 0 , . . . , 0 ) to the plane x 0 = 1 gives the reaIization of the hyperbolic space inside the sphera. These geometrical facts will present us with the possibility to include the affine and the hyperbolic R a d o n transforms in a unified construction. 1. B a s i c d e f i n i t i o n s . Let f ( x ) 9 O ( - n + 2). Then we define the conformal R a d o n t r a n s f o r m as the integrals of f on the sections of Q by the h y p e r p l a n e s L~ = {(~, x) = 0):
(3)
?(,)
=s
=
= (t0,''',fn+i)
s f(x),(<,, x>) 9 R n + 2 \ { 0}
Here we need to know f in a neighborhood of Q for the first version of the definition and only on Q for the second one. Of course ] is nothing more t h a n the projective
92 R a d o n transform of the distribution f(x)5([~(x)) but we will see t h a t it deserves the special consideration. It affine coordinates we have
(3')
/ ( ~ ) = --~
. I(X)~((~, X))CO(~,I,...,Xn+I )
Correspondingly
f({) = ~
(3")
~(x0,...,,,)=
,, f(x)5(({, X))We(XO,...,Xn),
Z (-1)J-'~,xJA<', O<_j<_n
~0=l,~j=-l,j>0
ir
Let us r e m a r k t h a t in b o t h cases we have forms w(x), w~(x) which are invariant relative to the groups SO(n + 1) and SO(n; 1) correspondingly. In the case of the sphera we consider integrals over all spheras of the codimension 1 - the h y p e r p l a n e sections. Between them are the big spheras corresponding to the sections through (0) ({0 = 0). So the conformal Radon transform extends not only the afflne R a d o n transform but also the Minkovski-Funk transform. On [Rn we have the o p e r a t o r of integration on hyperplanes and hyperspheras. We have the overdeteminant problem of integral geometry: functions of n variables on Q are transformed in functions of n + 1 variables. So we can expect t h a t the image is described by one equation. Indeed we can check directly t h a t functions from the image satisfy the wave equation
We need only to a p p l y the wave o p e r a t o r to (3) and use t h a t t a(t) = 0. We will prove t h a t conversely each O ( - 1 ) - s o l u t i o n of (4) belongs to the image. For the p r o o f we will construct the inverse o p e r a t o r out of the solutions of (4) to the functions on Q and our constructions will follow the conception of the o p e r a t o r ,~. Let us t a l k a b o u t the domain for f ( { ) . We will consider it for { E E where
(5)
= = {r # (0); r ~
.
.
~
.
.
.
.
2 1 < 0}. ~n-}-
For such ~ the sections have the dimension n - 1. Of course we could be put f - 0 outside E b u t it is not convenient for the reasons which we will outline below. 2. T h e o p e r a t o r ~. We want to reconstruct f through f . We will not consider the triviM possibility of reconstructing f ( x ) by the contracting of sections to the point x. If F E ( 9 ( - 1 ) on ~ we want to construct a n-form ~ F which will be closed if F is a solution of the wave equation. It is not difficult to guess the formula for such a o p e r a t o r ~. It was written down directly in [GRS], [G6]. Goncharov [Gon] found the formula using his general conception of the o p e r a t o r ~ on the language of :D-modules. So let us
aF[x] = a0F[z] - xlF[x] . . . . .
[~ -
<~7~> oF(~) o~---~-'
gn+lF[x]
+ n - ~ <~, ~ - io>--, ~(~)'
~j(~) = d~J~(r
93 It is convenient sometimes to use a variant of this formula:
i<j
t < x , - ~ - - i O ) n + n - 1 <x, ~ - - i 0 ) - l J
kr
d~k,
5ij = eixi(j - ejxj(i,
(6')
Dij = ei{j ~"~ii - ej{i ~--~i, 1 eo= l, gj = - l , j >0. This form depends on the point x 6 Q as on the p a r a m e t e r and it can be pushed down on E = E / ( N + ) x . Its coefficients are distributions which are regularized according to the recipe from Lecture l ( w i t h the extension to the complex domain). W h e n we will restrict this n-form on different n-cycles P C ~ we must be sure t h a t we can realize the regularization remaining in the limits of P. T h e manifold ~ can be realized as a spherical belt. T h e basic fact a b o u t the o p e r a t o r ~ is t h a t if F is a solution of the wave equation (4) ( in p a r t i c u l a r F = ] ) then the form sF[x] is d o s e d for all x 6 Q. It is a quite direct c o m p u t a t i o n using the Euler formula for homogeneous functions which you can consider as a exercise or you can look at computations in [G6]. 3. I n v e r s i o n f o r m u l a s . The problem of the inversion of the conformal R a d o n transform we u n d e r s t a n d as the problem of representating solutions F of the wave equation in the form F = f . We will integrate the closed form ~F[x] on different n-cycles P C s dim ~. = n + 1, for the reconstruction of f(x) and we want to prove that
fr gf[x] =
(8)
c(F)
f(x).
T h e n-dimensional homology group of E is one-dimensional a n d we can choose as its representative the section P of E by the h y p e r p l a n e (9)
~0 = 0
which p a r a m e t r i z e s the sections of Q through (0). On this cycle the forms for j > 0 and
gjF = 0
~F[x]lr = ~0F[x]lr We want to prove (8) for this cycle. We can use for the regularization of the restriction of tcF[z] on P any plane r with ~0 = 0 (notations of Lecture 1). Following the definition (6) we will decompose n0 on 2 terms ~ and ~ ' and will integrate t h e m separately. If the function f is affine even:
f(xo,
xl,...,x,)
= f(xo,-xt,...,-xn),
then
](~0, ~1,-.., ~n) : /(~0, -- ~1,-'-, --en);
94 therefore the integral of x~[x] on F is equal zero and the integral of x~[x] gives the inversion formula for the Minkovski-Funk transform on the sphera S '~. If f is odd then opposite the first integral is zero and the second one reconstructs f ( x ) ( it can be prove the same way as for the Minkovski-Funk transform but it will follow also from our results below). For general functions we obtain (8). This reconstruction uses not only ]Iv but also ]~lr. So for the inversion of the Minkovski-Funk transform for all(not only even) functions we need know not only integrals along big spheras but also derivatives along parallel sections. The same situation will be true for any cycle F C ~, non homological to zero: we can reconstruct the function if we know the integrals on all sections parametrized by the cycle and some transversal derivatives. Between these cycles will be cycles (10)
V~ -- {~; (Y, ~) = 0}
(sections passing through y) if D(y) > 0. The cycle which we considered is Fv, y =
(1, 0,...,0). Remarks. 1. We can interpret this result in the language of the boundary problems for the wave equation. If we know a solution F as well as some transversal derivatives on a cycle F then we represent F as ](~), ~ E F, and we can extend F as the solution F = ] on all ~ E ~=. So we solve the Cauchy problem with the data on F. If we could avoid using the derivatives (F is admissible) then we would solve the Goursat problem with the data on F. 2. For generic function f the form n ] has singularities on the b o u n d a r y of ~. It is essential: otherwise if we could be extend the form on S n+l we could not be reconstruct f because n-dimensional homology of S ~+1 are trivial. 3. Using the language of Lecture 1 we can interpret our formulas as the realization of f E O ( - n + 2) as cohomology from H ( n - 1 ) ( C Q \ Q, O ( - n + 2)). Here C Q is the complexification of the quadric Q and we consider cohomology of its complement to the real quadric Q. 4. For even n the operator n transforms into the differential operator and we have the local inversion formulas, 4. A d m i s s i b l e c y c l e s ( c o n n e c t i o n w i t h t h e p r o j e c t i v e R a d o n t r a n s f o r m ) . The cycle F is called admissible if nF[x]l r can be compute through F i r . We remarked that all cycles in ~. are not admissible. Let us consider the sections Fu (10) for y E Q , e.g. y0 = (1, 1, 0 , . . . , 0 ) . Then Fu0 is the section G0 + ~1 = 0 We have
[
(z0
(11)
=
-
;67-
1 -~ (n - 1)((x, ~) - i0) "-~
0](--~1,
~1,' 9',~n+l)] 0~1
w0(~).
95 In this form the differentiation on ~1 is tangent to Fy0 and we can see that this cycle is admissible but it is singular: it contains the boundary point ~0 = (1, - 1 , 0 , . . . , 0) of E and the form ~ ] will be in general singular at this point. However if a function f will be equal zero in a neighborhood of y0 then the form ~ ] admits the regular extension in a neighborhood of ~0 outside of E. Therefore on this class of functions the cycle Fro becomes regular and we have
~F ~][X] =2j~ F (X0 --Xl)](--~I, ~l,...,~n+l) ,o
,o
((x, ~) - iO)"
(12) w0(~) ----
E
(-1)J~J A d~i,
l~_j~n+l
4(2~)"
c = (_l)n/2( n-
wo(~) = c f ( x )
1)!'
x #
y0.
i~j
The first equality results from integration by parts. As for the second one we use that under the condition on f the cycle Fv0 is homological to the cycles Fy C -=, [](y) > 0. However we can see the same directly. The formula (12) coincides with the projecive Radon inversion formula for ~ ( U l , . . . , an+l) which is defined out of the condition (xl - x0)c2(Xl - x0, x 2 , . . . , x , + l ) = f(x0, Xx,..., x n + l ) ,
x q Q.
Out of this condition ~ E C9(-n) can be define using x E Q for all u, Ul # 0. Such y correspond to points of Q different of the north pole N = y0. As result we have on P : the preferred affinization ul = 1 and it is natural to consider ~o as the function on the atone space Ux = 1 . These constructions correspond to the stereographic projection out of N. We have
(la)
~(6,.-., ~,+1) = ] ( - 6 , 6, ~ , . . . , ~,+1)
= /p,, ~P(Yl,. 99, YnT1)~(Yl~I
+ ' ' " + Yn+l~n+l)og(Y) 9
As result we obtain (12) as the consequence of the Radon inversion formula and we have also the new proof of the conformal inversion formula (8) because ~ f is the closed form and it is enough to check (8) for one cycle. W h a t about the regularization of the form g ] on cycles Fv, y E Q, it is convenient to take ~ corresponding to the tangent hyperplane to Q at y. In our example ~ = (1, - 1 , 0 , . . . , 0) and corresponds to the preferred affinization at P~ Let us emphasize that we suppose that f is equal zero in a neighborhood of N and as the result ~2 is finite. It is not surprising that we can compare functions on the projective space P " and the conformal space C " only on their joint part - the affine space ~[n.
5. Other admissible cycles and the hyperbolic horospherical Radon transform. There are other admissible cycles which are singular for arbitrary f but can be transform in regular ones for functions which are equal zero in some domains on
96 Q. The plan is always the same: we verify that a cycle is formally admissible and then select conditions on functions which guarantee the appropriate extension of ~ / o u t s i d e E. Let F n, 7/= (0, 0 , . . . , 0, 1), be the cone (14)
~02 _ ~12 . . . .
~2 = 0
with the vertex in r/. Points of the cone parameterize such hyperpl~nes that their intersections with the hyperplane (~/, x) = 0 are tangent to Q. After the stereographic projection we have the family of spheras tangenting to a fixed sphera (the image of the section by L~ = {(r/, x} = 0}). Let us restrict n F on F~: (15) ~F[xl]r, =
E
E ejg}F[x] + (n - 1)((x, 14) - i0) n - ' l<j
D,~Fks~i,j A d4k,
1< j ~ n
where Dij and ej from (6') and a~ is the first term in g(6). To obtain this formula we need to remark that on F ~ only differential monomials containing d~n+l can be different of zero. Since the differential operators Dij are tangent to F n this cycle is admissible. The cycle F" belongs to the closure of ~ and intersects the boundary on (16)
F ' N {~n+l = 0}.
If the function f is equal zero in a neighborhood of the section by L , then we can extend g F outside ~ in the order to include (16) and we can reconstruct such functions by the integration on F u. For the regularization of the form it is natural to take the a~nization ( = ~7. H we were to remove from Q a neighborhood of the section by Lu we will obtain 2 connected components and it is natural to work with functions on only one out of them. Of course it is possible to consider the cone F n with any r/, [](r/) < 0. For the model H~ of the hyperbolic space the cycle F ~ parametrizes sections by hyperplanes which are parallel to generators of the asymptotic cone of the hyperboloid H~_. This family is invariant relative to the group of hyperbolic motions SO(n; 1) and the sections are horospheras. In this way we obtained the inversion of the horospherical Radon transform. It is instructive that we need to put some restrictions on the behavior of functions around the infinity L~ N Q. There are other examples of admissible cycles which are defining by some tangency conditions with submanifolds on Q of different dimensions [G6]. Moreover it is possible to describe all admissible cycles. 6 . T h e h y p e r b o l i c g e o d e s i c R a d o n t r a n s f o r m , If [B(y ~ > 0 (a point outside of the ball) then the essential part of Fy0 lies outside 7~ (some hyperplanes passing y0 do not intersect Q). The form ~J? will be for general f singular on these cycle. Nevertheless for a special class of functions we can to contract the integration on them to the integration on cycles in ~. Let ~/0 will be the polar for y0 relatively Q. It means that the points of the section of Q by the plane L(~ ~ = {(7/~ x} = 0} are the tangent points of the
97 tangent hyperplanes to Q through y0. Let us L(q ~ divides Q into 2 open parts Q• Let (17)
s u p p f C Q+
Then we can extend a / o u t s i d e E such a way that the cycle Fy0 will be homotopic in this extended domain to cycles in E and
(18)
~r g'][x] = 2c f ( x ) xO
We used (17) to reduce g to g'. For the the regulaxization we can use any plane passing through y0. We can to write the inversion formula if instead the condition (17) we suppose that f is zero in a neighborhood of L(q~ For the hyperbolic geometry we take y0 = (0, . . . , 0, 1). In affine coordinates (xn+l = 1) it will the point (0) ~md the plane L(r] ~ will be the hyperplane at infinity. We consider finite functions f on H~_ and the cycle Fx0 of hyperplanes through (0). This family is SO(n; 1)-invaxiant and sections axe hyperbolic hyperplanes. So (18) gives the inversion formula for the hyperbolic geodesic Radon transform. As the affinization ~ for the regulaxization usually it is taken t be the hyperplane through (0) invariant relative to the isotropy subgroup of the point x. We could see in the consideration of this example that for inversion formulas it is important not only to construct the operator n out of the image into closed forms but also to investigate carefully appropriate cycles. In this example there was a big invaxiance group but it was not really crucial. The construction of the operator ,r can be generalized in the case when instead of the quadratic form [3(x) we consider either another quadratic form or any homogeneous polynomial of any degree [G6]. However the choice of cycles is much more complicate in the more general situation and moreover appropriate cycles often do not existed. REFERENCES [FG]
[GG]
[GGG] laGS] [GGV] [G1]
[a2] [a3]
Faraut J., Gindikin S., Psedo-Hermitian Symmetric spaces of tube type, Topics in Geometry: In Memory of Joseph D'Atri (Gindikin S., ed.), Birkhauser, 1996, pp. 123-154. Gelfand I, Gindikin S., Nonloeal inversion formulas in real integral geometry, Funct.Anal. Prilozh. 11 (1977), no. 3, 12-19 (Russian); Engl.transl.: Funct.Anal.Appl. 11 (1977), 173-179. Gelfand I., Gindikin S., Graev M., Integral geometry in aj~ne and projective spaces,, Itogi nauki i techniki 16 (1980), VINITI , pages 55-224 (Russian); Engl.transl.: J.Sov. Math. 18 (1980), 39-67. Gelfand I., Graev M., Shapiro Z., Integral geometry on k-dimensional planes,, Funct.Anal. Prilozh. 1 (1967), no. 1, 15-31 (Russian); Engl.transl.: Funet.Anal.Appl. I (1967), 14-27. Gelfand I., Graev M., Vilenkin N., Integral geometry and representation theory,, Generalized Functions, vol. 5, Fizmatgiz, 1962 (Russian); Engl.transl.: Academic Press, 1966. Gindikin S., The Radon transform from cohomological point of view, 75 years of Radon transform (S.Gindikin, P.Michor, eds.), International Press, 1994, pp. 123-128. Gindikin S., Holomorphic language for O-cohomology and representations of real semisimpie Lie groups, Contemporary Math. 154 (1993), 103-115. Gindikin S., Integral formulas and integral geometry for "O-cohomology in CP n, Funct. Anal. Prilozh. 18 (1984), no. 2, 26-33 (Russian); Engl.transl.: Funct.Anal.Appl. 18 (1984).
98 [G41
[G5] [G6] [GH]
[GRS] [Gon]
Gindikin S., Unitary representations of groups of authomorphisms of Riemann symmetric spaces of null curvature, Funct.Anal. Prilozh. 1 (1967), 26-33 (Russian); Engl.transl.: Funct.Anal.Appl. 1 (1967), 28-32. Gindikin S., Fourier transform and Hardy spaces of-O-cohomology in tube domains, C.R. Aca~l.Sci.Paris 415 s~rie I (1992), 1139-1143. Gindikin S., Integral geometry on real quadrics, Amer.Math. Soc.Transl.(2) 169 (1995), 23-31. Gindikin S, Khenkin G., The Cauchy -Fantappie formula on projective space, Amer.Math. Soe.Transl.(2) 146 (1990), 23-32. Gindikin S., Reeds J., Shepp L., Spherical tomography and spherical integral geometry, Lectures in Applied Math. 30 (1994), Amer.Math.Soc., 83-92. Goncharov A., Integral geometry and W-modules, Math. Research Letters 2 (1995), 415435.
DEPARTMENT OF MATHEMATICS, HILL CENTER, RUTGERS UNIVERSITY, NEW BRUNSWICK, NJ 08903, U.S.A.
E-mail address: [email protected]
CIME Lectures Venice June 1996
RADON
TRANSFORMS
AND
WAVE EQUATIONS.
SIGURDUR HELGASON
1. I n t r o d u c t i o n . These lectures give a short account of the basic Radon transform theory on Euclidean and symmetric spaces. The applications are focussed on the wave equation with proofs. These applications are a mixture of old and new results, as described specifically in the text. Background material can, to a large extent, be found in my books [H9], [Hll] where some of the basic tools axe discussed in greater detail. I would like to thank the organizers, E.C. Tarabusi, M. Picardello and G. Zampieri, for inviting me to present these lectures. The following standard notation will be used. In a metric space X with metric d, C(X) denotes the space of continuous complex-valued functions, Br(x) denotes the open ball {y E X : d(x, y) < r} in X. For a manifold X, we use Schwartz' notation C(X) (resp. :D(X)) and T)'(X) (resp. C'(X)) for the space of C ~176 functions (resp. C ~176 functions of compact support) and distributions (resp. distributions of compact support). If X = R " , S ( R n) denotes the space of f c C(R '~) for which each derivative goes to 0 at oc faster than any negative power of the distance Ix[. For Lie groups the Lie algebra is denoted with the corresponding German letter, say G and 9. The adjoint representation of G is denoted Ad and of 9 by ad. 2. T h e F l a t Case. W a v e E q u a t i o n s . Let p n denote the set of hyperplanes in Euclidean space R% The Radon transform on R n assigns to each function f on R n, integrable over each hyperplane, the function f" on p n given by
(2.1)
:: ~ f(x)dra(x),
~ E pn,
where dm denotes the Lebesgue measure on ~. Along with the transform f ~ f" we consider the dual transform ~ ~ ~ which to a function ~ on P'~ assigns the function ~ on R n given by (2.2)
~(x) =: /
~(~)d#(~),
x E R n,
100
where d# denotes the average on the (compact) set of hyperplanes passing through X.
Representing a hyperplane ~ by the equation (x, w) = p where ( , 1 denotes the inner product, w a unit vector, and p E R, we see that P~ has a natural manifold structure. Since the pairs (w,p) and ( - w , - p ) represent the same ~, the mapping (w,p) ) ~ of S n-1 • R onto P~ is a double covering. Note that (2.2) can also be written
1/
~o(x) = -~
(2.3)
(p(w, (x,w) )d.w,
S,~-x
where dw is the volume element on S ~-1 and ~ its area. The inversion of the transform (2.1) can be expressed in various ways; for us the following formulation will be convenient. In analogy with the space $ ( R n) we consider the space S ( P n) of smooth functions on P " for which each derivative is rapidly decreasing in the p variable. We define the operator A by d~_i d--~=-r~(W,p),
(2.4)
do-,
7-lpd~=T~(w,p),
n odd n even
where 7-/denotes the Hilbert transform
(~F)(t) = ~i JR tF(P) - p dp, F e S ( R ) . Then (2.1) is inverted by (cf. [H2], ILl)
cf = (A f ) v
(2.5)
f e S ( R n)
where the constant c equals c = (-47r) ~
r(") r( 89
v
The transform T ~ ~ can be inverted similarly ([H4], [Hll], [Go], [So]) but this will not be needed here. As is well known, the Radon transform is particularly well suited to a study of the wave equation 02U
(2.6)
=
= f0(x),
=
where L = f i 0~ (0~ = 0/0x~) and fo, fl E 73(R"). Our treatment resembles t h a t i=l
of Lax and Phillips [LP1], CoL 2.1, p. 105 for n odd (with some simplifications) but applies to b o t h cases n odd and n even.
101 Lemma
If h C C2(lq.) and w C S n-l, then the function
2.1.
v(x, t) = h((x,w) + t) satisfies Lv = (Ot)2v. The proof is obvious. For a function ~ on S n-1 • R we now put 0~
0~ = ~ ,
Theorem
(2.7)
(n~)(~,p) = np(~(~,;)).
The equation (2.6) has the solution
2.2.
(0 ~ - ' ^fo + c ~ - 2 ~ ) (w, (w, x +) t) (dw n
~ ( z , t ) = cl
odd)
s,~-,
u(x,t)=c2
/
(7"t(On-lffo+On-2~))(w,(w,x}+t)dw
(neven)
s,~- i The constants cl, c2 are given by Cl = 89(21r) l - n , c2 = 89(27ri) 1-n
PROOF. (i) n odd. Because of Lemma 2.1 the right hand side does satisfy the wave equation so we just have to verify the initial data. For this note that the function ~
(0n-~s
(w, (w, x})is odd so u(x,0) = f0(x) follows from (2.5). Applying 0t
raises the derivatives in (2.7) by one and now the function w ~ (Onfo)(w, (w, x)) is odd so again ut(x, O) = fl(x) by virtue of (2.5). (ii) n even. Here we just have to use the commutation 07-t = ?-/0; otherwise the proof is the same as for the case n odd. It is also easy to write down the solution to (2.6) using the Fourier transform
Y(~) = f f(x)e-i(~'r . 2
It"
In fact, assuming the function x ~
(2.s)
u(x, t) in S ( R n) for a given t we get
~,,(r t) + (~, ~)~(~, t) = 0
and solving this ordinary differential equation we obtain (2.9)
~(~, t) = f0(~) cos ]~]t + j~ (~)sin [~]t
102
The function ( ) (sin [(It)/l(I is entire of exponential type Itl on C '~ (of at most polynomial growth on R n) so by the Paley-Wiener theorem on R ~ there exists a Tt E E ' ( R '~) such that sin [~lt - f/ e-i(r
(2.10)
.
R~
The following result is suggested by (2.9) but is readily verified even if fo and fl do not have compact support. Theorem
2.3.
Given fo, fl E $ ( R n) the function
(2.11)
u(x, t) = fo * T[ + fl * Tt
satisfies equation (2.6). Here T[ stands for Ot(Tt) and * denotes convolution.
C o r o l l a r y 2.4. If fo, fl have support in BR(O) then u has support in the region
]xl < Itl + R. In fact, the support of Tt is contained in Bltl(0 ) - so formula (2.11) shows that the function x , u(x, t) has support in (BR+Itl (0))-. Combining this with Theorem 2.2 we can deduce another classical result. C o r o l l a r y 2.5. (Huygens' principle). Let n > 1 be odd. A s s u m e fo and f l have support in BR(O). Then u has support in the shell (2.12)
Itl - R < Ixl ___ Itl + R
which is the union (2.13)
[_J
C~
IyI<_R
where Cy is the light cone
c~ = { ( x , t ) :
I x - yl = Itl}.
To verify Corollary 2.5 note that since n is odd, Theorem 2.2 implies (2.14)
u(0, t) = 0 for Itl > n.
If z E R '~, F C E ( R '~) we denote by F z the translated function y ~ F ( y + z). Then u z satisfies (2.6) with initial data f~),f~ which have support contained in BR+IzI(0). Hence by (2.14) uZ(O,t) = 0 for It] > R + ]zl, i.e. (2.15)
u(z, t) = 0 for Itl > n + Izl.
In view of Cor. 2.4 this proves (2.12).
103 Finally, if ( x , t ) 9 Cy with lYl -< R we have I x - y l : [tl so Ix[ _< Ix-yI+lyl <_ ]tl+R and It] = I x - Yl-< Ix[ + R, proving (2.12). Conversely, if ( x , t ) satisfies (2.12) t h e n t ] ~x = K[(] x x ] - I t l ) which has n ~ (x,t) 9 < R" R e m a r k 2.6. principle ([CH]).
~2
If c ~ 0 the o p e r a t o r L + c - ~
does not satisfy Huygens'
3. Huygens' Principle and Conformal Maps. Let M be a homogeneous space with a Lorentzian structure ( p s e u d o - R i e m a n n i a n s t r u c t u r e of signature (+, , , . . . , - ) . Let D be an invariant differential o p e r a t o r on M . T h e light cone Co t h r o u g h the origin o E M is m a d e up of the isotropic geodesics t h r o u g h o. A fundamental solution for D is a d i s t r i b u t i o n S E :D~(M) satisfying D S = 5 where 5 is the delta distribution at o. D is said to satisfy Huygens' Principle if such an S exist with s u p p o r t contained in Co. In general one considers this only in a neighborhood of o in M which is the diffeomorphic image of a n e i g h b o r h o o d of 0 in M0 under the exponential m a p p i n g Exp. Since a conformal diffeomorphism ~ maps isotropic geodesics into isotropic geodesics t h e image of a Huygensian differential o p e r a t o r D under ~ is again Huygensian [(O)]. We can now make this more explicit for the Laplace-Beltrami operator. Let M1 and 21//2 be two Lorentzian manifolds of dimension n and T:
M1----* M2
a conformal diffeomorphism, i.e. T'g2 = T2 gl, where T is a nowhere vanishing function on M1. If K 1 , / ( 2 are the scalar curvatures on M1, M2 respectively and ca = (n - 2 ) / 4 ( n - 1) we have the formula for the Laplacians L1 a n d L2, (3.1)
(L1 - cnK1) ( ' r ~ - l ( f o T ) )
= T ~+1
( (L2 - c n K n ) f ) o T
([Co], [H10], p.332). In particular, the null spaces of t h e o p e r a t o r s L1 - cnK1 and L2 - cnK2 correspond under T. T h e same being the case with the light cones we conclude t h a t L1 - cnK1 satisfies Huygens' principle if and only if L2 - cnK2 does. Let us now recall R a d o n ' s general problem: D e t e r m i n e a function from its integrals over certain submanifolds. We consider a special case of this problem. Let X = G / H be a homogeneous space, H being a closed subgroup of the Lie group G. For g C G we p u t H g = g H g - 1 . By a generalized sphere we m e a n an orbit H g 9 x x being a fixed point in X . If X is a Euclidean space or more generally a two-point homogeneous space and G its group of isometrics, a generalized sphere is an o r d i n a r y sphere. T h e orbital integral problem ([H1]) amounts to the d e t e r m i n a t i o n of a function on X in terms of its integrals over generalized spheres.
104
Let us consider the cases when G / H is a Lorentzian manifolds of constant curvature. Up to local isometry and normalization these are (the subscript 0 denoting identity component) X ~ = RnO0(1,n-
1)/O0(1,n-
X - = G / H = O0(1, n ) / O 0 ( 1 , n X + = 0o(2, n-
1)/Oo(1,n-
1) -
flat case
1)
curvature - 1,
1)
curvature + 1.
For these cases a function can be determined on the basis of its integrals over generalized spheres ([H1]). Here we sketch the result for the middle case X - = G / H , G = O 0 ( 1 , n ) , H = O0(1, n - 1). For a substantial generalization see [Or]. Here X - is identified with the quadric
x~-x~
(3.2)
.....
x 2n + l = - 1
with t h e Lorentzian structure induced by the quadratic form (3.3)
x~
x~
e
. . . . . .
Xn+
1
and H t h e subgroup of G fixing the point o = ( 0 , . . . , 0, 1). Select a bi-invariant measure dh on the u n i m o d u l a r group H. Let u 6 ~D(X-), g C G a n d x~ at a (Riemannian) distance r > 0 from o. We can then define the o p e r a t o r M ~ (the orbital integral) by (3.4)
1 u ( g h - x ~ ) d h - A(r)
(M~u)(g 9 o) = H
/
u(z)dw~(z),
Sr(g'o)
where S~(g. o) is t h e orbit Hg "gxr, dw~ the R i e m a n n i a n measure on St(g" o) a n d A(r) a function of r alone. The measure dh can be normalized such t h a t A(r) = ( s i n h r ) n-1 (for X ~ and X + we would have A(r) -- r n - l , ( s i n r ) n - l , respectively). In R i e m a n n i a n geometry, lim S~(o) = o
7"---*0
lim M~ u = cou,
r~0
where co is a constant. In the Lorentzian case lira S~(o) = Co,
r~O
where Co is a semicone with vertex o. W h a t a b o u t lim M~u? Here we have the r ---~0
following result ([H1] and [H9], Ch. I, w Theorem
3.1.
Assume n = d i m X -
even, n = 2m, and put
Q(L) = (L + ( n - 3)2)(L + ( n - 5 ) 4 ) . . . (L + l ( n -
2))
105 Then if u E ~(X-),
(3.5)
u ( y ) : c lim r n - 2 ( M r Q ( L ) u ) ( y ) r--*0
whe c = - 2)! Formula (3.5) solves the orbital integral problem for the space X - . By carring out the limit in (3.5) the following consequence is obtained in [SS]:. C o r o l l a r y 3.2. 6 = Q(L), w h e r e v is a n H - i n v a r i a n t m e a s u r e o n Co - {o}.
As remarked at the beginning of this section this means that Q(L), and in fact each factor L k = L + ( n - k ) ( k - 1) in Q(L), satisfies Huygens' principle. For the "middle factor" L + m ( m - 1) this could have been predicted from (3.1) as follows. The space X - is conformally equivalent to X ~ (see [O]) and has scalar curvature - n ( n - 1). Formula (3.1) thus relates the operators L + m ( m 1) and the fiat wave operator L ~ (the Laplacian on X ~ so, by the above mentioned conformal invariance of Huygens' principle, L + m ( m - 1) does indeed satisfy it. The fact that all the operators Lk (k = 3, 5 , . . . , n - 1) satisfy Huygens' principle provides an interesting contrast with Remark 2.6.
4. The Hyperbolic Space. Now let X denote the hyperbolic space H '~ of n-dimensions. We use as a model the unit ball Ixl < 1 in R '~ with the Riemannian structure 2
(4.1)
(1
-
2
2 xl
.....
22
~
which induces the distance function d. The k-dimensional totally geodesic submanifolds ~ are the k-spherical caps perpendicular to the boundary Ix I = 1. Let denote the manifold of all these ~. The Radon transform is here given by
(4.2)
7(r
= //(x)dm(x),
r e
=
v
d m being the Riemannian measure on ~. The dual transform ~ ----* ~ maps func-
tions on ~: to functions on X by (4.3)
~(x) = /
~o(()d#(~),
the average of ~ over the ~ passing through x. The transform f inverted as follows ([H1]).
f" can be
106
T h e o r e m 4.1.
For k even let Qk(z) denote the polynomial
Qk(z) = [z + ( k - 1 ) ( n - k)][z + ( k - 3 ) ( n - k + 2)]... [z + 1. ( n - 2)] of degree k/2. Then f ~
"f is inverted by the formula cf = Qk(L)((f) v)
(4.4)
where L is the Laplaee-Beltrami operator and
c- r~(-4~)k/2. For k not necessarily even the inversion is more complicated. For this we generv alize the dual transform as follows. Let p > 0 and define the transform ~ ) ~p by
d(x,~) =p
the average over the ~ at distance p from x. For k arbitrary the inversion formula (4.4) is replaced by the following ([H10]),
(4.5)
f ( x ) = co
(.)(x)(u 2 - v 2
- t dv
0
u=l
where im(v) = cosh-l(v -1) and 2k+1 co
- - ( k -- 1 ) ! ~ k + 1
~k+l being the area of S k. For the transform f theorem ([H2], [H7]). T h e o r e m 4.2. (4.6)
'
) l o n e has the following support
Suppose f C C ( X ) satisfies the decay condition:
For each m E Z +, x ~
f ( x ) e md(~
is bounded.
Then if f(~)=O
for
d(o,~) > A
f(x)=O
for
d(o,x) > A.
we have
The proof uses the fact that the geodesic spheres for the metric (4.1) are ordinary spheres and secondly the metric (4.1) is conformal to the flat metric on R n. This second property was used by Kurusa [K] to relate the transform f ~ f" to the flat
107
k-plane transform on R n. Such a relationship can be used to describe the range for the transform f ----* f" (see [BT], [I] and for k = n - 1 [Gi]). C o r o l l a r y 4.3.
The X-ray transform on X is injective on the space of functions
satis/ying (4.6). This holds also for the symmetric spaces X = G / K considered in the next section ([H8], p. 133). 5. R a d o n T r a n s f o r m o n a S y m m e t r i c S p a c e . Let X = G / K be a symmetric space of the noncompact type, i.e. G is a connected noncompact semisimple Lie group with finite center and K a maximal compact subgroup. Let g and ~, denote their respective Lie algebras. Let o denote the origin e K in X. We consider the Iwasawa decompositions G = N A K , g = n + a + ~ where N and A are nilpotent and abelian respectively. We write accordingly g = n(g)expA(g)k(g),
A(g) e a.
A horocycle ~ in X is an orbit Ng 9 x of some point x E X under a conjugate Ng = gNg -1 of N. Let -E denote the set of all horocycles ~. The group G acts transitively on ~ and the subgroup which maps the horocycle ~0 -- N . o into itself equals M N , where M is the centralizer of A in K. Thus (5.1)
-E = G / M N = ( K / M ) • A.
The Radon transform on X maps functions f on X into functions f on -E by (5.2)
f'(~) -- j~ f ( x ) d m ( x ) ,
~e
where dm is the Riemannian measure on ~ up to a constant factor (independent of ~). In group-theoretic terms, (5.3)
f'(g" ~0) = / f ( g n . o)dn
(g 9 G)
N
and here the Haar measure dn on N is normalized by the relation e2p(A(~
= 1
N
where 9 is the Cartan involution of G with fixed point set K and (5.4)
2p(H) = T r ( a d g l n ) ,
g 9 a.
Since ~ = ( K / M ) • A we sometimes write r
~o) = ~(kM, a)
108
if ~ is a function on ~. The dual transform ~ ~ (5.5)
~(x) = f
V .
~ is given by
~(~)dp(~)
the average over the space of horocycles passing through x. terms,
(5.6)
. o) = [
In group-theoretic
(gk . o)dk.
K
To put this in a form analogous to (2.3) we introduce a vector-valued analog of (x, w} in (2.3). For x = gg 9 G/K and b = kM in K/M we put
A(x, b) = A(k-lg) 9 a.
(5.7) Then ([H5])
(5.8)
~(x) = /
~(kexpA(x, kM).~o)e2p(A(x'kM))dkM
K/M
where
K/M
dkM is the normalized K-invariant measure on K/M. We also write B for db for dkM. For A 9 a*, the complex dual of a, we consider the spherical
and function
(fix(g) = /
e (ix+p)(A(kg))dk.
K
If a + C a is the positive Weyl chamber and a*+ the corresponding chamber in a* it follows from [HC], p. 291 (see [Hb], Ch. IV, w that the limit (5.9)
c(A) =
lim e(-ix+p)(tH)9)j~(ex p t H )
t~+co
exists provided Re(i;~) 9 a*+ and the limit is independent of H 9 a +. From the formula for this c-function ([GK]) in terms of G a m m a functions it follows that it extends to a meromorphic function on a* and it is easy to prove that 1/c(A) has each of its derivatives bounded by a polynomial on a* ([H3]). Consider now the Euclidean Fourier transform on A (5.10)
F*(),) =
f F(a)e- x( ~
.X 9 a*.
The measures da and dA are normalized such that the inversion formulas hold without a multiplicative constant. We then define the operators j and j on ,S(A) by
(iF)* (:,) = c(A)-I F* (:9 (jF)* (:~) = c ( - ~ ) - 1F* (~)
109
and the operators J and J by (J~)(kaMN) = ja(~(kaMN)) ( ] ~ ) ( k a M g ) = ~a(~(kaMN))
the subscript a denoting the variable on which j acts. Finally we put A = e - P J o eP, A = e - p ] o eP where eP(kM, a) = ep0~ The transform f ~ f" is inverted by the following result. ([H2], [H5], [Hll]).
Theorem 5.1.
With A and A defined as above
(5.11)
f = w-l(hAf) v
f 9 7)(Z)
where w is the order of the Weyl group W of X .
For G complex another inversion is given in ([GG]). The operators A and .~ are differential operators precisely when G has all its Cartan subgroups conjugate. R a n g e s a n d kernels. The Radon transform f ---* f can be extended to all f 9 L I ( X ) and then it is still injective [H5]. The range :D(X) ^ can also be described explicitly ([H6]) and it implies the surjectivity, E(E) v = E(X) for the v dual transform ~ ~ ~. However this last transform has a huge kernel so there is no useful inversion formula for it (in contrast to R n) except under the condition of K-invarianee ([Hll], V, w E x a m p l e . To illustrate the description of the range mentioned for the map f ~ 7 and the kernel of ~ ~ ~v we consider the simplest case X = H2 (without the factor 4 in (4.1). The horocycles are the circles internally tangent to Ix I = 1. Let ~t,e denote the horocycle through eie, with non-Euclidean distance t (with sign) from 0. A function r 9 E(~) is determined by its Fourier series
(5.12)
r
= E
Cn(t)e~n~
r
9 E(R).
nEZ
Theorem 5.2. (i) The range D(H2) A consists of the functions (5.12) such that for each n C Z, (5.13)
Ca(t) = e-t ( ~ t - 1 ) ( d _ where ~
3)... (d-
2In] + 1 ) ~ n ( t )
E Z)(R) is even. v
(ii) The kernel of ~ --~ ~ on E(E) consists of the functions (5.12) such that for each n C Z,
(5.14)
(d+l)
(d+3)...(d+2,n,-1)etr
isodd.
110
R e m a r k . For X = R 2 the analogous result is the following: Let ~t,o denote the line (x, e i~ = t and expand a function r on p2 by (5.12). Then r E :D(R 2) if and only if for each n, (5.15)
dlnl = d--~Tn(t),
r
T~ E D ( R ) even
and r E E ( P 2) is in the kernel of the dual transform if and only if for each n,
dtnl dtin ICn(t)
(5.16)
is odd.
As a part of the range characterization of D ( X ) ^ one has the following support theorem ([H6]). T h e o r e m 5.3. Let B C X be a closed ball. If f E 7P(X) and ff(() = O for MB=O then f(x)=O forx• B. This still holds with f sufficiently rapidly decreasing. The proof in [H6] is based on studying K-types; a different proof was given by [GQ]. 6.
The Fourier Transform on X = G/K.
T h e Fourier transform on R n can be written
f(Aw) = / f(x)e-i'~(x'~)dx. R=
To define a Fourier transform on X = G/K we replace (x, w) by the vector-valued inner product A(x, b) in (5.7). Thus we put for a function f on X,
f(A, b) = f f(x)e(-i;~+P)(A(x'b))dx
(6.1)
X
for all A E a~, b E B for which the integral converges. Here dx = dgK is the volume element on X, defined by the Haar measures dg and dk. The measure dg is normalized by
f f(g)d9= f G
f(nak) e-2p('~
NAK
in the notation of w Theorem (6.2)
6.1.
The Fourier transform is inverted by
1 / f(x) -----w
f(A,b)e (i;~+p)(A(x'b)) dAdb
ic(A)] 2 ,
f E D(X).
a*xB
Moreover, f ~
f extends to an isometry of L2(X) onto L2(a~_x B; ]c()~)]-2dA db).
111
If f is K - i n v a r i a n t this reduces to H a r i s h - C h a n d r a ' s m a i n results for the spherical transform on X which is the s y m m e t r i c space analog of t h e Bessel t r a n s f o r m (X)
F(%) = j F(r)Jo(rA)dr. 0
For applications to differential equations it is useful to have a P a l e y - W i e n e r t y p e theorem, t h a t is an intrinsic characterization of the space T)(X)':. Here the result is as follows ([H6]). T h e o r e m 6.2. The range T)(X)~ is given by the space of/unctions ~ E $(a* • B) satisfying a) /~ ~ ~(%, b) is an entire function of exponential type, uniformly in b. b) The integral
/
~(,1, b)e(i~+P)(A(~'b))db is W-invariant in )~.
B
7. T h e W a v e E q u a t i o n o n X = G / K . Denoting the Laplace-Beltrami o p e r a t o r on X by L x we consider t h e shifted wave equation
(Lx + Ipl 2) ~ -- ~Ot2
(7.1) with Cauchy d a t a (7.2)
u(x,O) : fo(x), ut(x,O) = fl(x),
f o , f l E l)(X).
T h e shift by Ipl 2 is related to t h e curvature t e r m in (3.1). In this section we assume r a n k X : 1. Let H0 be the unit vector in t h e positive Wely chamber a +. Since the c-function c(,~) is singular at ,~ = 0, the o p e r a t o r s J and J can b e factored by
J = HoJ',
J = H0J',
where J ' and J ' are new operators on functions on E. Theorem
The solution to the Cauchy problem (7.1)-(7.2) is given by
7.1.
1
^ B
Remark.
For X -- H 3 this agrees with formula 3.41 in [LP2].
112
We prove this by means of a couple of lemmas.
L e m m a 7.2. Let F E C2(A), b C B. Then the function v(x, t) = F(exp(A(x, b) + tHo))e -Iplt
(7.4)
satisfies (7.1). PROOF. Because of the K-invariance of the equation (7.1) we may assume b = eM. Then both (7.4) and (7.1) are N-invariant so it suffices to verify (7.1) on A . o . The restriction of L x + [p[2 to N-invariant functions is on A 9 o given by the N-radiM part A N ( L x + Ipl 2) which equals ePLA o e-P (see [H9], Ch. II) where LA is the Laplacian on A- o. In our case, LA = H3 so the problem is to verify
)
(ePH2 o e-P)(F(aexp(tHo)))e -Iplt = Or-~ F(aexp(tHo))e -Iplt . Using p(Ho) = IPl this is a routine matter. L e m m a 7.3.
(7.5)
For f E •(X)
we have the identity
/ (J1])(eP f)(b, exp A(x, b))ep(A(='b))db = O. B
The same identity holds with j i j replaced by HoJ]. PROOF. First recall the relationship between the Radon- and Fourier transforms, valid for X of any rank, (7.6)
f(A, b) ---- f(ePf')(b, a ) e -i)~(l~
a)da
A SO
y(A, b) 2
= f J'2'(ePf)(b' a)e-i~O~ A
Using the invariance condition in Theorem 6.2, b) for x = 0 we deduce that
(J'J') (ePf) (b, a)db B
is even in a and consequently,
/ J'](ePf~(b, e)db = 0
B
113
being the derivative at a = e of an even function on A. This proves the lemma for x = o. On the other hand, A = e - p J o e p,/~ = e-P J o e p and if we define A ~ -- e - P J ~ o eP the expression in the lemma is ( A ' A f ' ) v (x). However, the operators ^,v,~., A' are all G-invariant ([H5], p 42) so since the expression (7.5) vanishes for x = o it vanishes identically. This proves the lemma since the statement for H o J J follows in the same way. PROOF OF THEOREM 7.1.
In the integrand in (7.3) we replace e p(A(z,b)) by eP( A(x,b)+tHo) e-lPlt"
Then for b fixed the integrand has the form (7.4) so the right hand side of (7.3) certainly satisfies the wave equation (7.1). It remains to check the initial conditions. We put t = 0 in the right hand side of (7.3). Then the term containing fl vanishes by virtue of L e m m a 7.3. Introducing A = e - P J o e p, A = e - p ] o e p and making use of (5.8) we see from the inversion formula (Theorem 5.1) that the right hand side of (7.3) reduces to f o ( x ) . Hence u ( x , O) = f o ( x ) . For the initial condition u t ( x , O) = f l ( x ) we apply (OlOt)o to (7.3), apply L e m m a 7.3 for H o J J and deduce that the term in (7.3) containing f0 gives no contribution to u t ( x , O ) . Again we deduce u t ( x , O) = f l ( x ) from the inversion formula. This proves Theorem 7.1. C o r o l l a r y 7.4.
The solution to the Cauchy problem ( 7 . 1 ) - ( 7 . 2 ) can be w r i t t e n
(7.7) K
In fact, if g = e this agrees with formula (7.3) (with x = o). By the G-invariance of (7.1) a translate by g C G in the functions f0, fl implies a similar translate in the solution u. Since the operators f ---o f', A,/~, A ~ are all G-invariant formula (7.7) holds in general. Remark. Theorem 2.2 could also be stated in the form of (7.7) but then we would have to consider the orientation preserving isometry group of R n as acting on S '~-1 • R , not on p n . This difference between R n and G / K of rank 1 can be traced to the fact that our hyperplanes in R n are not given an orientation whereas for G / K we always have two horocycles passing through a given point with the same tangent plane. Other quite different explicit solution formulas for the wave equation (mostly in terms of spherical averages) can be found in [P], [BO], JIM] for X of rank 1 (using the classification) and in [CY] and [Hll]V, w for more general X. In fact the solution to (7.1), (7.2) for X of any rank can be written ([Hll], V, w
(7.8)
u ( x , 0 = (/0 x E;)(x) + (/1 x &)(x)
114
where gt E g ' ( X ) satisfies (7.9)
(Lx + Ipl ~) s~ = ~-~t~s~
(7.10)
go = 0, E~ = 60
g~ denoting a g t. Moreover g.t has Fourier transform given by ~(A, b) - sin(IA[t)
and thus has support in the ball Bltl(O)-. In (7.8) x denotes the convolution on X induced by the convolution product on G. H u y g e n s ' P r i n c i p l e . The operators J, J are differential operators exactly when c(A) -1 is a polynomial. For X of rank 1 this happens only for the odd-dimensional hyperbolic spaces H ~. C o r o l l a r y 7.5. Let X = H 2n+l. Suppose fo and fl in (7.2) have support in the ball B~(O). Then the solution u to (7.1)-(7.2) is supported in the conical shell
Itl
- R <_ d(o, x) <_ It[ + R.
In fact our solution formula implies immediately that (7.11)
u(o,t) = 0
for Itl > R.
Next let g E G and ug(x,t) = u(g -1. x,t), f[(x) = f i ( g - 1 , x). Then ug satisfies (7.1)-(7.2) with initial data fg which have support in Bn(g.o)- C (Bn+d(o,g.o)(o))-. Hence by (7.11), ug(o,t) = 0 for It[ > R + d(o,g, o) whence (7.12)
u(y,t)=O
if
]tl>_R+d(o,y ).
On the other hand (7.8) shows that x ---* u(x,t) has support in BR+ltl(O). This and (7.12) proves the corollary. 8. T h e M u l t i t e m p o r a l W a v e e q u a t i o n o n S y m m e t r i c Spaces. We consider again the symmetric space X = G / K and the algebra D ( G / K ) of G-invariant differential operators on X. From [HC] w we consider the isomorphism F : D ( G / K ) ---* I(a) where I(a) is the set of W-invariants in the symmetric algebra S(a). Given p E S(a) we let O(p) denote the corresponding constant coefficient differential operator on a. Each c0(p) operates on the symmetric algebra S(a*). Let H(a*) C S(a*) be the subspace (the space of harmonic polynomials) annihilated by all a(p), p E I(a), without constant term. Then dimH(a*) = w ([HC], w
115
Let pl = 1 , p 2 , . . . ,pw be a real homogeneous basis of H(a). We consider now the following system for a function u 9 E(X • a): (8.1)
Du = 0 ( r ( D ) ) ~
for an D e D ( G / K )
with initial data
(8.2)
(O(pr
=fi(x),
1
Here fl,.., fw are given functions in 7)(X). Note that in (8.1) D acts on the first argument and in (8.1)-(8.2) 0(F(D)) acts on the second. We think of the elements of a as multidimensional time so (8.2) is indeed an initial condition. In the case when X has rank 1 D ( G / K ) consists of the polynomials in L x and F ( / x ) = LA --Ipl 2. Also Pl -- 1,p2 = H so the system (8.1), (8.2) reduces to (7.1),
(7.2). The system (8.1), (8.2) was first studied in [STS] and later in [S] and [PS]. We shall write down the solution of the system in a new form. T h e n using the Radon transform we shall rewrite the solution in a more geometric formulation. For more details see [H12] and [HS]. Consider the quotient field C(S(a)) and a bilinear form ( , > on it with values in
a"b"
(a, b) = Z
a, b e C(S(a))
a6W
where a~(;Q = a(a-1)~) ()~ 9 a*). Define qJ by (8.3)
(qJ,Pi) = 5ij,
1 <_i,j < w.
T h e n the qJ are rational functions on a* but the denominators are relatively simple; in fact ([HC],w
7rqj 9 S(a),
(8.4)
where ~T is the product of the indivisible positive roots. Writing by (8.4) qJ(),) -- hJ(A)/Ir(s where hJ 9 S(a) the sum
qJ(ia)~)e ia~'(s)
(H 6 a)
a6W
equals (since ~r(io)~) = e(o)v:(i)~) where le(a)l = 1)
(8.5)
-1 Z a6W
The sum is skew and is thus divisible by the polynomial ~r(iA). The expression in (8.5) is therefore a W-invaxiant entire function on a c of exponential type < IHI of at most polyomial growth in A. By the Paley-Wiener theorem (6.2) or rather
116
its distributional version ([H6], Theorem 8.5, or [EHO]) there exist distributions S~ E E'(X) such that
(8.6)
q~(i~)e i~(") = f a6W
e(-i~§
.
X
Actually the distributions S~4 are K-invariant and have support in BIH ](o) ([Hll], p. 281). T h e o r e m 8.1.
Given f l , . . . , f w E E(X) the system (8.1) (8.2) has a solution
given by w
u(x, H) = E ( f j
(8.7)
• SJH)(X).
j=l
SKETCH OF PROOF. We have if S c E'(X),
D ( f • S) = f • DS.
(8.8)
The definition of S~4 shows easily that (8.9)
(DS~A,
b) = r(D)(iA)(S~4 ~.
Next we apply the operator O(F(D)) to (8.6) in the H-variable. Because of the Winvariance of F(D) this amounts to multiplication by F(D)(iA). Comparing with (8.9) this proves
DSJH = O(F(D)H(S~),
(8.10)
n E D(G/K).
This, combined with (8.8), shows that u satisfies
n u = O(F(n))u.
(8.11)
To verify (8.2) we apply O(Pk)H to (8.6). Then
E
qj (ia)~)Pk(ia)~)ei~'~(H) = / e(-i~+P)(A(='b))d(O(Pk)S~)"
aEW
X
Putting H -- 0 and using (8.3) we get
(8.12)
{o(pk)s . }.=o
=
where 50 is the delta function at the origin of X. Since f x 50 = f the initial conditions (8.2) now follow immediately from (8.12) and (8.7).
117
R e m a r k . Formulas (8.7), (8.10) and (8.12) are the exact analogs of (7.8), (7.9) and (7.10) respectively. C o r o l l a r y 8.2.
/f supp(fj) C BR(O) for 1 < j <_w then for each H C a
u(x,H) = O for x ~ BR+tHl(O). This is an immediate consequence of formula (8.7). Thinking of H as time one can express this as "the speed of propagation" being <_ 1. This appears as Theorem 3.4 in [PS] with a rather complicated justification. We shall now rewrite (8.7) in terms of the Radon transform. Since S~ is K invariant the Fourier transform of fj > S3H is the product of the individual Fourier transforms ([Hll], III w which are fj(A, b) and the left hand side of (8.6). Hence by the inversion formula (Theorem 6.1) we have
u(x,H) =
1/
j
"/1}
le( )l 2
a*xB
Because of the invariance in Theorem 6.2 b) this can be written
E / ep(A(x'5)) /
qJ(iA)
~(~ b)ei~(H+A(x,b))d)~ db.
e(~)e(-~) J ~ ' ' ' ' 3
B
a*
Now let Jj denote the operator on A which corresponds to the multiplication by qJ(i)Q/Ic()Q[ 2 on a* under the Fourier transform on A and put Aj = e-PJj o ep. Using (7.6) and the G-invariance of Aj we then obtain the following result. T h e o r e m 8.3.
Let f l , . . . , f w C I)(X).
Then the solution u in (8.7) can be
written (8.13)
u(x,H) = / ~ Jj(ePL)(b, exp(H + A(x,b))ep(A(x'b))db, B
j=l
and also (8.14)
o(. o,.) : . . ( - , / g K
(..o.p. ,o).. 1
The analogy with Theorem 7.1 is striking. We can also derive the following analog of Cor. 7.4. This corollary was proved in IS] in a different way. C o r o l l a r y 8.4. Assume that all the Cartan subgroups of G are conjugate. Then the following Huygens' principle holds:
118
Ifsupp(fj) C BR(o) for 1 < j < w then the solution u(x, H) in (8.7) has support in the region
{(x,H) : I n l -
(8.15)
R <_ d(x,o) < R +
Inl}.
PROOF. Using (8.4) and Cor. 6.15 in [H9], Ch. IV we see that q3(iA)/Ic(~)l 2 is a polynomial. Formula (8.13) implies
u(o,H) = 0
if
IHI
~
R
because each Jj is now a differential operator and fj(b, a) = 0 for all b e B if I logal _> R. More generally, u(x, H) = 0 if I n + A(x, b)l > R for all b C B. Now if x = gK, b = K M g = nakl we have ([Hll], IV, w (13)).
IA(x,b)l = IA(k-lg)l _< d(o,x) Thus if
IHI >
for all b.
R + d(o, x) we have IH + A(x,b)l >_ I H I - IA(x,b)l >_ I H I - d ( o , x ) > R
so u(x, H ) -- 0. This shows the left half of (8.15). The right half is a consequence of Cor. 8.2. Formula (8.14) is of course a generalization of (7.7). Lemma (7.2) also generalizes as follows. P r o p o s i t i o n 8.5.
(8.16)
Let F E C2(A), b E B. Then the function v(x, H) = F (exp(A(x, b) + H) ) e -p(H)
is a solution of the system (8.1). Again it suffices to prove this for b = eM and then the function v is N-invariant. If D e D ( G / K ) then A N (D), the N-radial part of D is given by AN (D) = ePP(D)o e-P; then the statement for v is easily verified. We conclude with a result kindly communicated to me by Rouvi6re and published here with his permission.
Proposition 8.6. For each f E T)(X) the function (8.17)
F(gK, H) = ep(H) f
f ( g k e x p H n , o)dk dn
KxN
is a solution of the system (8.1). This follows from the transmutation theorem ([H3], [Hll]) (8.18)
AD~ = F(D)A~
119
for the Abel transform
(.4~o)(a) = ep(l~
S ~o(an. o)dn N
(~0 e :D(X) K-invariant) combined with the Darboux equation ([H1], [Hll])
(8.19)
With the notation f~(x) = f f(k. x)dk the right hand side of (8.17) is K
eO(~oga)/ ( / ~ - l ) ~ ( a
n
.
o)dn
(a = expH)
N
which by applying F(D)~ gives by (8.18), (8.20)
ep(loga) / D(fg-1)~ (an. o)dn. N
However, using (8.19),
so, substituting into (8.20), the result follows.
REFERENCES
[BO] U. Bunke and M. Olbrich, The wave kernel for the Laplacian on classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function, Ann. Glob. Anal. Geom. 12 (1994), 357-405. [BT] C. Berenstein and C. Casadio Tarabusi, Range of the k-dimensional Radon transform in real hyperbolic spaces, Forum Math. 5 (1993), 603-616. [CH] R. Courant and Hilbert, Methoden der Mathematischen Physik, vol. II (Springer, eds.), Berlin, 1973. [Co] E. Cotton, Sur les invariant differentiels ..., Ann. l~c. Norm. Sup. 17 (1900), 211-244. [cv] O.A. Chalykh and A.P. Veselov, Integrability and Huygens' principle on symmetric spaces, Preprint (1995). [EHO] M. Eguchi, M. Hashizume and K. Okamoto, The Paley- Wiener theorem for distributions on symmetric spaces, Hiroshima Math. J. 3 (1973), 109-120.
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I.M. Gelfand and M.I. Graev, The geometry of homogeneous spaces, group representations in homogeneous spaces and questions in integral geometry related to them, Amer. Math. Soc. Transl. 37 (1964). [Gi] S.G. Gindikin, Integral geometry on quadrics, Amer. Math. Soc. Transl. (2) 169 (1995), 23-31. [GK] S.G. Gindikin and F.I. Karpelevic, Plancherel measure of Riemannian symmetric spaces of non-positive curvature, Dokl. Al~d. Nauk USSR 145 (1962), 252-255. [Go] F. Gonzalez, Radon transforms on Grassmann manifolds, J. Funct. Anal. 71 (1987), 339-362. [GQ] F. Gonzalez and E.T. Quinto, Support theorems for Radon transforms on higher rank symetric spaces, Proc. Amer. Math. Soc. 122 (1994), 1045-1052. [HC] Harish-Chandra, Spherical functions on a semisimple Lie group I, Amer. J. Math. 80 (1958), 241-310. [H1] S. Helgason, Differential operators on homogeneous spaces, Acta Math. 102 (1959), 239-299. [H2] _ _ , A duality in integral geometry; some generalizations of the Radon transform, Bull. Amer. Math. Soc. 70 (1964), 435-446. [H3] _ _ , Fundamental solutions of invariant differential operators on symmetric spaces, Amer. J. Math. 86 (1964), 565-601. [Hd] _ _ , The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153180. [H5] _ _ , A duality for symmetric spaces with applications to group representations, Advan. Math. 5 (1970), 1-154. [H6] _ _ , The surjectivity of invariant differential operators on symmetric spaces, Ann. of Math. 98 (1973), 451-480. [H7] _ _ , Support of Radon transforms, Advan. Math. 38 (1980), 91-100. [HS] _ _ , The Radon Transform, Birkhs Boston, 1980. [H9] _ _ , Groups and Geometric Analysis; Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York, 1984. [H10] _ _ , The totally geodesic transform on constant curvature spaces, Amer. Math. Soc. 113 (1990), 141-149. [Hll] _ _ , Geometric Analysis on Symmetric Spaces, Amer. Math. Soc. (1994), Math. Surveys and Monographs. [H12] _ _ , Integral geometry and multitemporal wave equations, Comm. Pure Appl. Math ((to appear)). [HS] S. Helgason and H. Schlichtkrull, The Paley- Wiener space for the multitemporal wave equation, Comm. Pure Appl. Math. ((to appear)). [I] S. Ishil~wa, The range characterizations of the totally geodesic Radon transform on the real hyperbolic space, Preprint, Univ. of Tokyo, 1995. [IM] A. Intissar et M. Val Ould Moustapha, Solution explicite de l'dquation des ondes dans l'espace symdtrique de type non compact de rang 1, C.R. Acad. Sci. Paris 321 (1995), 77-80. [K] A. Kurusa, Support theorems for the totally geodesic Radon transform on constant curvature spaces, Proc. Amer. Math. Soc. 122 (1994), 429-435.
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[L] [LP1] [LP2] [O1
[Or] [P]
[PS] [S] [So] [ss]
[STS]
D. Ludwig, The Radon transform on Euclidean space, Comm. Pure. Appl. Math. 23 (1966), 49-81. P. Lax and R.S. Phillips, Scattering Theory, Academic Press, New York, 1967. __, Translation representations for the solution of the non-Euclidean wave equation, Comm. Pure. Appl. Math. 32 (1979), 617-667. B. Orsted, The conformal invariance of Huygens' principle, J. Differential Geom. 16 (1981), 1-9. J. Orloff, Orbital integrals on symmetric spaces, in 'Noncommutative Harmonic Analysis and Lie Groups', Lecture Notes in Math. 1243 (1987), 198239. E. Pedon, Equations des ondes sur les espaces hyperboliques, Preprint, Univ. de Nancy (1992-93). R.S. Phillips and M. Shahshahani, Scattering theory for symmetric spaces of the noncompact type, Duke Math. J. 72 (1993), 1-29. M. Shahshahani, Invariant hyperbolic systems on symmetric spaces, Differential Geometry (R. Brooks et al., eds.), Birkhs Boston, 1983, pp. 203233. D.C. Solomon, Asymptotic formulas for the dual Radon transform, Math. Zeitschr. 195 (1987), 321-343. R. Schimming and H. Schlichtkrull, Helmholtz operators on harmonic manifolds, Acta Math. 173 (1994), 235-258. M.A. Semenov-Tian-Shanski, Harmonic analysis on Riemannian symmetric spaces of negative curvature and scattering theory, Math. USSR Izvestija 10 (1976), 535-563.
ANALYTIC
DISCS AND
EXTENDIBILITY
THE
OF C R F U N C T I O N S
ALEXANDER TUMANOV
One of the most striking fundamental phenomena in several complex variables is the forced extendibility. By the classical Hartogs theorem of 1906, if D is a domain with connected boundary bD in C N, N _> 2, then all hmctions holomorphic in a neighborhood of bD extend to be holomorphic in D. For a domain D C C N with smooth connected boundary bD, Bochner (1943) gave a proof of this theorem for CR functions. The latter are functions on bD that satisfy the Cauchy-Riemann equation in directions tangential to bD. In these lectures we describe recent results on the extendibility of CR functions on manifolds of higher codimension. Our ultimate goal is to prove that CR functions on a minimal CR manifold extend to a wedge. The exposition here is self contained. In particular, we cover the Bishop equation, the Baouendi-Treves approximation theorem, and the edge-of-the-wedge theorem by Ayrapetian and Henkin. I would like to thank CIME and the organizers of the Venice session in June 1996 for the pleasure to lecture there.
124
I. C R m a n i f o l d s and C R functions In these lectures we do n o t worry a b o u t the regularity questions. We always assume t h a t all manifolds, hmctions, etc, are as s m o o t h as we need. Since we are interested in local problems, by a manifold (e.g., a hypersurface) we always m e a n a locally
closed manifold, t h a t is a closed submanifold in an open set of C N. We first consider real hypersurfaces in complex space. Let M C C N be a s m o o t h real hypersurface. T h e tangent space Tv(M ) to M at p 9 M is a real h y p e r p l a n e in
Tp(C N) -~ C N. Let Tv(M ) be the m a x i m a l complex subspace in Tp(M). T h e n Tp(M) = Tp(M) N JTp(M), where J : C N ~ C N is the o p e r a t o r of multiplication by i = ~L--f. T h e complex
tangent spaces T~(M) form the complex t a n g e n t bundle TO(M). Locally, M can be defined by t h e equation r ( z ) = 0, where r : C N --~ ]R is a s m o o t h function with dr ~ O. T h e n
Tp(M) = {~ 9 c N : (dr(p),~) = 0}. We have dr -- Or + -Or, where
Or = E r ' d z "
= E rjd-2 j ~
rj = Or~Oz,,
r: = O r / ~ j .
Then
Tp(M) = {~ 9 c N : (Or(p),~) = 0}. A C 1 function f on M is called a CR function if df is C-linear on T ~ ( M ) , p E M . In other words, f is a C R function if
df A dzl A . . . A
dZN[M =
O.
This condition uses only holomorphic differentials dzj = dx i +idyj. For a continuous function f on M , we say t h a t f is a C R function if the above condition holds in the sense of distributions. We are interested whether all C R functions on a hypersurface M e x t e n d locally to be holomorphic in a one-sided neighborhood of p C M .
T h e celebrated Hans
Lewy t h e o r e m answers the question in terms of the Levi form of M at p. Recall t h a t t h e Levi form of M is a h e r m i t i a n form on T~(M) defined b y N
L(p)(G~) = E j,k=l
where rj~ = 02r/Ozy O-hk.
rJk (p)~j~k,
~ 9 Tp(M),
125
T h e o r e m 1.1. (H. Lewy, 1956) Suppose L(p)(~,~) # 0 for some ~ 9 T~(M). Then all CR functions on M extend holomorphically to the same one-sided neighborhood ofp 9 M . More specifically, let L(p)(~,~) > O. Then all CR functions on M extend locally to the side r(z) < 0 near p. Tr~preau [Tr] found a necessary and sufficient condition for the one-sided extendibility. Theorem
1.2. (Tr6preau [Tr]) Let M be a hypersurface in C N and let p 9 M.
Assume there is no complex hypersurface S C M passing through p. Then all CR functions on M holomorphicaUy extend to the same one-sided neighborhood of p. Conversely, if such a hypersurface S exists, then for every neighborhood U C M of p there is a CR function f in U that does not extend to any one-sided neighborhood ofp. We will o b t a i n these theorems as special cases of a more general result. We now consider manifolds of higher codimension.
Let M be a s m o o t h real
submanifold in C N. We define the complex tangent space at p 9 M the same way as for hypersurfaces:
Tp(M) -- Tp(M) M JTp(M), p 9 M. T h e manifold M is called a CR manifold if dim T~(M) does not d e p e n d on p E M . T h e manifold M is called generic if Tp(M) spans Tp(C N) -
C N over C for all
p E M , t h a t is
Tp(M) + JTp(M) = C g. For example, all real hypersurfaces are generic. If M is generic, then M is a C R manifold a n d d i m e Tp(M) + codim M = N, where c o d i m M is t h e codimension of M in C N. T h e dimension d i m c T ~ ( M ) is called t h e C R dimension of M and is denoted by C R d i m M . W e define C R functions on M the same way as for hypersurfaces: a C 1 function f is a C R function if df is C-linear on Tp(M) for all p E M or equivalently, df A
dZl A . . . A dzg [M of distributions.
= 0. We a p p l y this definition to continuous functions in t h e sense
126 2. T h e B a o u e n d i - T r e v e s
approximation
theorem
T h e celebrated Baouendi-Treves theorem states t h a t a C R function locally can be a p p r o x i m a t e d by holomorphic polynomials.
Theorem
2.1. (Baouendi and Treves [BT]) Let M be a generic manifold in C N.
Then for every point p E M there is a neighborhood U C M of p such that for every continuous CR function f on M there is a sequence of polynomials f~ such that f;~lv converge uniformly to f l u as ~ --* oo. Proof. Let Mo be a m a x i m a l l y real submanifold t h r o u g h p, t h a t is p E Mo, TC(Mo) = 0, a n d d i m Mo = n. We introduce coordinates in C y in such a way t h a t p = 0 a n d
Tp(Mo) -= R N C C N. Shrinking M0 if necessary, we assume t h a t there exists 0 < a < 1 so t h a t for all distinct z, w E M0, we have
(2.1)
IIm (z - w)[ < ~ IRe (z - w)l.
We introduce t h e entire functions
f~(z) = ()~/Tr)g/2 f
f ( w ) e - ~ ( z - ~ ) 2 d w i A . . . A dWN,
JM o
where (z - w) 2 : = ~ ( z j - wy) 2, ,~ > 0. The condition (2.1) ensures t h a t
(.k/z~)g/2e-)~(z-w)2dWl A ' " A dWN form a 5-shaped sequence as ,k --* exp. Thus
f~(z)~f(z)
for
z9
as
,~-~co.
We prove t h a t f~ --~ f in a neighborhood of p 9 M. Let us view M0 as a manifold with boundary, a n d let M1 be a slight p e r t u r b a t i o n of Mo with the same boundary. ~Ve define ]~ = f M : ' ' "
by integrating the same expression as in f~. T h e n by
the same argument, ];~(z) ---* f ( z ) for z 9 M1. But actually ]~(z) = f~(z) for all
z E C g . Indeed, M0 and M i b o u n d a submanifold Mol C M , OMoi = Mo - Mi. Since e -~(z-~)2 is holomorphic and df A dwi A ... A d w g l M -~ 0, the integrand is a closed form on M . By the Stokes formula f~ - s
= fMo -- fM: = fMol = 0. Thus
f~ converge to f on every p e r t u r b a t i o n M i of Mo, hence in a neighborhood of p on M . To a p p r o x i m a t e f by polynomials, one takes the Taylor polynomials of f~. T h e t h e o r e m is proved.
127
3. Bishop's equation Let A C C b e t h e s t a n d a r d unit disc A _-- {~ C C : I~1 < 1}, and let bA = {~ C C : I~1 = 1} be the unit circle. A n analytic disc in C g is a m a p A : A --* C N holomorphic in A a n d s m o o t h up to the boundary. Let M b e a manifold in C N. We say t h a t the disc A is a t t a c h e d to M if A m a p s the circle bA to M . Let f b e a C R function on M and let A be a sufficiently small analytic disc a t t a c h e d to M . T h e n f is a limit of a sequence of polynomials f~ t h a t converges to f on bA := A ( b A ) c M . Since the functions f ~ o A converge on bA, by the m a x i m u m principle, t h e y converge inside A. Thus f~ converge on the set A ( ~ ) . a set filled up by small discs a t t a c h e d to M .
Let ~ be
T h e n the sequence of holomorphic
functions f), converge uniformly on ~t. If ~ is an open set in C N, then the limit of f~ is holomorphic. If ~t is a C R manifold, then the limit is a C R function. In 1965 E. Bishop [Bi] introduced a construction of analytic discs a t t a c h e d to a real manifold M c C g.
Proposition 3.1. Let M be a CR manifold in C~N = C n • C "~ and let p E M . Assume that the projection w : C g --~ C n maps T ~ ( M ) isomorphicalIy to C n. Then for every q C M close to p and every sufficiently small analytic disc r : A ---* C n with r
= w(q) there exists a unique analytic disc A attached to M such that
woA=r
andA(1)=q.
Proof. Let p -- 0.
We choose w to be p a r t of coordinate functions in C N a n d
complete it to a system of holomorphic coordinates (z = x + iy, w) E C " • C n. We first assume M is generic. T h e n we can choose the z coordinates so t h a t T~(M) has t h e equation x = 0, so T p ( M ) has the equation z = 0. T h e n M has a
local equation x = h(y, w),
where h is a s m o o t h function with h(0) = 0 and dh(O) = O. Let ~ ~-* A(~) = (x(~) + iy(~), w(~)) be an analytic disc in C g . T h e disc A is a t t a c h e d to M if a n d only if A(~) E M for ~ E bA. T h a t is (3.1)
x(0 = h(Y(0, w(0),
M = 1.
Since x(~) a n d y(~) are harmonic conjugates, they are related by t h e Hilbert t r a n s f o r m on bA, t h a t is y = T x + c, where T is the Hilbert transform on t h e unit
128
circle bA, and c = ~-~
)dO = const. By (3.1) we have
(3.2)
y --- T h ( y , w) + c
The equation (3.2) is Bishop's equation. Given the analytic disc ~ ~-~ w(~) in C n and c E R TM, the solution ff ~-* y(ff) defines the disc A by harmonically extending z(~) = h(y(~), w(~)) + iy(~) from bA to A. Since our disc is specified by the condition A(1) -- q, we modify the equation in the following way. For a function ~b on bA, we set T1r := T r - (Tr
The
function T1r is characterized by the conditions 1)
r + i T1r extends holomorphieally to A;
2)
(T1r
= 0.
Then (3.2) turns into y = T l h ( y , w) + y0,
where Yo = y(1). Given ~ ~-~ w(ff) = r
and Y0 = (the y-component of q) E R TM,
the solution defines the disc A. The existence and uniqueness of the solution to Bishop's equation follow by estimates in the Lipschitz space Ck,~(k _> 1, 0 < a < 1) because the term T l h ( y , w) is small. (See [Bo] and [T2].) We now consider the case in which M is not generic. Then we split the z coordinates z = (z', z") so that the projection (z, w) --~ (z', w) takes M to a generic manifold M ~ C C g' and the projection M ~ M ~ is a C R diffeomorphism, t h a t is it maps T O ( M ) isomorphically to TO(Mr). The inverse mapping M ' ~ M is given by the C R function z ' . By the first part of this proof, there is a unique disc A ~ attached to M I with w-component ~ ~ r
and A'(1) = (z'(q), w(q)). By the Baouendi-Treves approx-
imation theorem, the C R function z" extends to be holomorphic inside the disc A 1. The extension gives the z" component of the needed disc A. The proof is complete. For future reference we include the following. C o r o l l a r y 3.2. Let M be a real not necessarily CR manifold in C N = C n • C m, and p E M .
A s s u m e that f o r every q E M the projection w : C N --~ C n maps
T q ( M ) injectively to C ~. Then f o r every q E M close to p and every sufficiently small analytic disc r
A --. C n with r
= w(q) there is at most one analytic disc
A attached to M such that w o A = r and A(1) = q.
129
Proof. Note that dim T~(M) is an upper semicontinuous function of q, that is it can only j u m p down at a nearby point. Thus, without a loss of generality we assume t h a t dimc T~(M) = n. As in the previous proof, we choose w to be part of coordinate functions in C N and complete it to a system of holomorphic coordinates (z, w) E C m x C a. We again split the z coordinates z = (z', z") so that the projection (z, w) --* (z', w) diffeomorphicaUy maps M to a generic manifold M t C C N'. Then if there were two discs attached to M with the indicated properties, then their projectons to C N' would violate Propostion 3.1. The proof is complete.
4. T h e e d g e - o f - t h e - w e d g e t h e o r e m Let M be a generic manifold in C N. Let Np(M) := Tp(CN)/Tp(M) be the normal space to M in C N. The spaces Np(M) form the normal bundle N ( M ) . Let F be an open cone in Np(M). Let U be a neighborhood of p in M. We can identify F with a cone in a transversal plane H through p, H @ Tp(M) = C N. A
wedge W with direction cone F is a set of the form
w= ((Mnu)+r)ng. T h e following version of the edge-of-the-wedge theorem is due to Ayrapetian and Henkin [AH].
T h e o r e m 4.1. Let M C C N be generic, p C M. Let Mj 1 < j
<m=codimM
be manifolds with boundary M, in particular, dim Mj -- dim M + 1. Let ~j C Tp(Mj)/Tp(M) C Np(M) point inside Mj and let ~j form a basis in Np(M). Let F :-- C o n v { ~ l , . . . , ( m } be the convex span of ~,...~,~. Then all continuous CR m functions on M U Uj=I Mj extend holomorphically to the same wedge with direction
cone F', where F' C F is any finer cone, that is F ' \ { 0 } C F. m
Proof. We will show t h a t analytic discs attached to X = M U Uj=I Mj fill up a wedge W with edge M and direction cone F I. Then by the Baouendi-Treves approximation theorem, all CR functions on X extend to be holomorphic in W, what we need. Let p = 0. We introduce local holomorphic coordinates (z, w) E C m x C n = C N and define M and My (1 < j _< m) by a parametric equation
x = h(y, w, t),
130
where t C ~ m , and h is a s m o o t h Rm-valued functoin in a n e g h b o r h o o d of 0 E Rm • C n • Rm. T h e n the manifold M has the equation x = h(y, w, 0) while M j is defined as tj > 0, tk = 0 for k ~ j . We choose h so t h a t h(0, 0, 0) = 0, where
by,
hy(0, 0, 0) = 0,
h~(0, 0, 0) = 0,
ht(O,O,O)=I,
h~, and ht denote the derivatives of h, and I is the identity matrix. In
these coordinates the cone F turns into ] ~
:= {t E ]~m : tj _> 0, 1 _< j _< m}.
m Mj. Let ( H A ( ( ) = (x(() + iy((), w(()) be a disc a t t a c h e d to X = M U ~Jj=l
Note t h a t (y, w, t) form a set of local coordinates in a neighborhood of 0 E C g . Let ~-* t ( ( ) be t h e t-component of the discs A in the coordinates (y, w , t ) . T h e fact t h a t A is a t t a c h e d to X is expressed as follows. Let t ( ( ) = ( t l ( ( ) , . . . , tm(()). T h e n all ty(() _> 0 for ](] = 1, b u t for every ( E bA only one of tj-s can differ from 0. m We divide the circle bA into disjoint arcs bA = Uj=I-~j a n d want A to m a p ~,j
to My. Let Cy be a s m o o t h function on bA such t h a t Cj(() > 0, Cy(() -- 0 outside yj. We define for ( E bA t(~) = (/~l(~l(~),...,)~rn~)m(~)),
where A = ( - ~ 1 , . . . , / ~ m ) C ]~m. We take w ( ( ) = wo = const and y(1) = Yo C R m. T h e n the disc A satisfies the Bishop equation (4.1)
y = Th(y, Wo, t) + Y0-
T h e disc A depends on p a r a m e t e r s Yo, wo and A C 1~'~. We identify (Yo, Wo) with the point (h(yo,wo,0) + iyo,wo) E M. The disc A is a t t a c h e d to X if and only if
We consider the evaluation m a p
F : M xR'~---+CN ~ M x R ~ F : (Y0, w0, ~) ~ A(0) = (x(0) + iyo, w0), the center of A. T h e m a p p i n g F is defined in a neighborhood of 0 E M • R "~. T h e set W = F ( M x R TM) consists of the centers of discs a t t a c h e d to X. We will show t h a t the derivative F~(0) of F as a m a p p i n g M x R "~ ~ M • R m has m a t r i x
131
where the asterisk denotes unimportant elements. Hence, W contains a wedge with direction cone F' for every cone F ' finer than F - R + , what we need. __
m
Note t h a t for A = 0, the solution of (4.1) is y - 0, so A(() = (x0 + iyo,wo) and
x(r = h(yo, To, 0). Thus F[Mx{O} = id[Mx{O}. For general A we have A(0) = (x(0) + iy(O), To), x(O) = Pox, where 1 f2~ P o e = ~ Jo r176
(4.2) is the mean value of r
We now find Fx = OF/OA. We have
Fx(O, O, O) = (Pox~, 0), AtA=0,
w0=0,
Poxa = Po(hyyx + hits).
yo=0wehavehy=0,
tA =
0)
hi=I, "..
0
,
r
so PoX~ = Pot~. We choose the Cj-s such that PoCj = 1. Then F~(O,O,O) = (I, 0). Thus F~(O) has the needed form, which completes the proof.
5. L e v i f o r m a n d t h e w e d g e - e x t e n d i b i l i t y The H. Lewy extension theorem has a generalization to manifolds of higher codimension. In the final form, the result is due to Ayrapetian and Henkin [AH] and Boggess and Polking (see [Bo]). Let M C C N be a generic manifold in C g. Locally M is given by a local equation r ( z ) = 0,
where r = (rz,. 9 9 rm) is a smooth Rm-valued function in a neighborhood of 0 E C N such t h a t Or1 A... AOrm ~ O. Let H(p)(X, Y) be the complex Hessian of r, t h a t is
N j,k=l
n(p)(X,7) = ~
C~2T Xj~kk E C m,
OzjO~k
where X , Y e T~(M) C C N. We identify Np(M) with ]~m by dr(p): Np(M) --* R m, the differential of the defining function. Let V : = C o n v { g ( p ) ( X , X ) : X e Tp(M)}. The cone F C Np(M) is called the Levi cone of M at p. One can show that this definition does not depend on the choice of the defining function r.
132
T h e o r e m 5.1. Let M be a generic man]old in C N, and let p 9 M . Suppose the Levi cone F C N p ( M ) has nonempty interior. Then for every finer cone F ~ C F, that is F'\(O} C F, there exists a neighborhood U of p in C N such that all CR ]unctions on M extend to be holomorphic in the wedge W = ( (U M M ) - F') M U with edge M and direction cone - F ~. Proof. We introduce coordinates (z = x + iy, w) E C TM x C n = C N so t h a t p = 0 and M has local equation
r = x-
h(y,w) = O,
where h(0) -- 0 a n d dh(O) -- O. T h e n Tp(M) and T~(M) are defined by the equatons x -- 0 a n d z = 0 respectively. Let a E T~(M) ~- C n. We will construct a one parameter family of discs ~ ~-~ A((, t) attached to M so t h a t the centers of the discs trace a smooth curve t ~-* A(0, t), t > 0 in the transversal plane II = {(z, w) : y = 0, w -- 0} --- Np(M). We will show t h a t the curve has direction vector
d t=oA(O,t ) dt
-2H(p)(a,E)
at p. Repeating this construction for all nearby points q E M yields a smooth family of curves. Let Ma be a u n i o n of these curves. T h e n ]Via is a smooth manifold with part of its b o u n d a r y in M. Since Me contains the curve through p, then
TpMa has
direction -H(p)(a,-5) 9 Np(M). T h e n the theorem will follow by Theorem 4.1, the edge of the wedge theorem applied to all Ma. We now construct the needed family ~ ~ A(~, t), t > 0 by Bishop's equation y = Th(y, w).
(5.1)
Since we want A(0) 9 YI-- {w = 0, y = 0}, we take c = y(0) -- 0 i n (3.2). We further take ~ ~-~ w(~) in the form
w(r = act, where a E C n --- Tp(M). T h e n A(0) = (x(0), 0) e II, and x(O) -- Pox, where P0 is defined by (4.2). The centers A(0) -- A(0, t) form a smooth curve. We now find its direction at t -- 0. We denote by the dot the derivative with respect to t. We have
J:(O) -- Po(hyy + h,,(v + h~w). Note t h a t for w ----0 the solution of (5.1) is y - 0, so at t -- 0 we have w -= 0, y -- 0, x - 0. Therefore hy = 0, hw -- 0, and hence 4(0) = 0 at t = 0. Thus A(0,0) = 0
133
and the direction of the curve t ~-~ A(0, t) is determined by the second derivative A(0, 0). We have
~/ --- T(hu~/ + h~o(v + h ~ ) . At t = 0 the expression after T is identically equal to 0, hence Z) - 0 at t -- 0. We have
2(0) = Po(hy~) + hyyyy + 2h~ywy + 2h~ywy + hwwzb~b + h~--~--~ + 2h~w~hw). At t = 0 the partial derivatives of h are evaluated at (0, 0, 0), so they are constant. Sincehy = 0and9
= 0 a t t = 0, then all terms w i t h y o r
~) are zero. We have
zb = a~, so the term hww(v(v is holomorphic in r and vanishes at ~ -- 0. Then its mean value is Po(h~ow~b~b) -- 0. The same applies to the antiholomorphic term. Recall r = x - h(y, w). We have
~(0) It=o = 2hw~a-d Po ( ~ ) = 2hwwa-d -= - 2 H ( p ) ( a , ~), so the curve has direction -2H(p)(a,~) at t = 0, what we need.
The proof is
complete. We recall the intrinsic definition of the Levi form. Let C T ( M ) = T ( M ) | C be the complexified tangent bundle to M and let T1,0(CN) the bundle of (1,0) vectors in C g.
CTC(M) = TI,~
+ T~
Then TI,~
where T~
L ( p ) ( X , Y- -) : = ~1[ X , Y](p)
~---
{~-~.j=laj_~..~j}N 0 be N) n C T ( M ) , and
= TI'~ = TI,~ We set mod CTp(M),
where X , Y
are sections of TI,~ One can show that this definition depends only on X(p), Y(p). The Levi form n(p) is a bilinear hermitian form on TpX'~ with values in (TB(M)/Tp(M)) | C. Note that J : C N ---* C N the multiplication by i = x/-L~ identifies T p ( M ) / T p ( M ) with Np(M). The relation between n(p) and H(p) the intrinsic and extrinsic definitions of the Levi form is given by (see [Bo]) dr(p)JL(p) = H(p). In view of the intrinsic Levi form, the condition that the Levi cone F has n o n e m p t y interior F is equivalent to the condition that the commutators [X, Y] (p) of sections of TI'~
span Tp(M).
We say that M has finite type k at p E M if the tangent space Tp(M) is spanned by the commutators
[... [xlE1 ,x2s ] , . . . , x k ~k ](p),
(5.2) where Xj are sections of
TI'~
is the ease of finite type 2.
O
and X; j is either Xj or X~. The ease of r # 0
134
6. Minimality Let M be a C R manifold in C N. We say t h a t M is minimal at p E M if there is no p r o p e r C R submanifold S C M through p such t h a t C R d i m S = C R d i m M . Since S is only locally closed in M , the word "proper" means t h a t dim S < dim M . T h e m i n i m a l i t y condition was first introduced in IT1]. Note if M has finite t y p e at p C M, then M is minimal at p. Indeed, if S C M with the i n d i c a t e d properties exists, then X j in the definition of finite t y p e can be restricted to S, the c o m m u t a t o r s (5.2) are in CTp(S), so t h e y cannot span the whole space Tp(M). In case M is real analytic, M is minimal at p if and only if M has finite t y p e at p. T h e m i n i m a l i t y is necessary and sufficient for wedge extendibility. Theorem
6.1. (Tumanov [T1]) Let M be a generic manifold in C N. Suppose M
is minimal at p C M. Then there is a wedge W with edge M near p such that all CR functions on M extend to be holomorphic in W. Theorem
6.2. (Baouendi and Rothschild [BR]) Let M be a generic manifold in
C N. Suppose M is not minimal at p C M. Then for every neighborhood U C M of p there exists a CR function in U that does not extend to any wedge with edge M near p. T h e o r e m s 6.1 and 6.2 generalize Tr@preau's Theorem 1.2. We will give a p r o o f of T h e o r e m 6.1. Let ylp denote the set of all discs A : ~ --~ C N a t t a c h e d to M through p E M, t h a t is A(1) = p. Let ~ denote the inner normal to the unit circle. To every A E ~4p we associate its n o r m a l direction at p: A H [A.(1)] c Np(M), where A~ = OA/Ov, a n d the brackets denote the class of a vector in Np(M). We set F = C o n v {[A~(1)] : A e Ap} C Np(M), the convex span of all Av(1) in Np(M). Lemma
6.3. Suppose that the cone F has nonempty interior. Then all CR func-
tions on M extend to the same wedge W with edge M near p. This l e m m a follows by the edge of the wedge theorem and the following
135
6.4. Let A c AB and ( = [A.(1)] r 0. Then all CR functions on M extend to be CR on a manifold 1~I such that 01VI = M near p and A~(1) E Tp(l~) points inside 1~I. Lemma
Pro@ Let ( z , w ) C C m • C n form a system of coordinates in C N so t h a t w : C TM x C n --~ C n maps T~(M) isomorphically to C n. Let p = 0. T h e n every small disc a t t a c h e d to M is uniquely defined by A(1) and its w-component ~ --* ~(~). By P r o p o s t i o n 3.1, for every/5 = (x0 +iyo, w0) close to p = 0, there is a disc A a t t a c h e d to M through/5, t h a t is A(1) =/5, with w-component ~ ~ @(~) = w(r + w 0 , where ~-* w(~) is the w-component of the given disc A. Let / ~ / = UA((0, 1)) the union of the radii of the discs A. T h e n - g / i s a s m o o t h manifold near p E M . Since 1~/contains A((0, 1)), t h e n A . ( 1 ) E Tp(f/I). Since 1~/ is covered by discs, t h e n all C R functions on M extend to/V/. T h e l e m m a follows. We will show t h a t if M is minimal at p C M , then the cone F has n o n e m p t y interior, and T h e o r e m 6.1 follows from L e m m a 6.3.
7. T h e d e f e c t o f a n a n a l y t i c d i s c We introduce a central notion in the proof of T h e o r e m 6.1, the defect of an analytic disc a t t a c h e d to a C R manifold. The original definition of the defect was given in [T1]. We give a more intuitive geometric definition of the defect due to Baouendi, Rothschild a n d T r @ r e a u [BRT]. Let M be a generic manifold in C N. Let T*I,~ or (1,0) forms in C.g.
Every co E T*I'~
coj C C. Note t h a t T*I'~
N) be the bundle of holomorphic
has the form co = ~cojdzj, where
N) ~ C N • C N is a complex manifold. The conormal
bundle N*(M) of M in C.N is the real dual bundle to N ( M ) . and holomorphic forms by co ~
Reco.
We identify real
In this identification N*(M) becomes a
submanifold in T * I ' ~
N;(M) =
{co e T ; I ' ~
RecolT,(M) = 0}.
Since T* 1,0 ( c N ) is a complex manifold, we can consider analytic discs in it. Let A be an analytic disc a t t a c h e d to M . We say t h a t an analytic disc A* : A --* T * l ' ~ is a lift of A to N* ( M ) if A* (~) E N~(r
N)
for ~ C bA. Let VA be the set of all
lifts of the disc A to N* (M). T h e set VA is a real vector space. We introduce the defect d e f A of t h e disc A as follows. def A = dim VA,
136
VA ----{ A * : A* is a lift of A to N * ( M ) } . Generally, N*(M) is not a C R manifold, so there might not be any nontrivial analytic discs a t t a c h e d to N*(M). Therefore, the existence of nontrivial lifts of A is r e g a r d e d as degeneracy of A, and d e f A measures the degree of this degeneracy. We set
VA(() = {A*(() : A* C VA},
( e bA.
It t u r n s out t h a t the spaces VA(() have the same dimension d e f A . L a m i n a 7.1. For every ( E bA the evaluation map VA --~ VA(() defined as A* ~-*
A*(() is bijective. Proof. Since M is generic, then the fibers Np(M) are t o t a l l y real in T ; I , ~
-~
C N, so t h e complex tangent spaces to N* (M) have dimension at most n -- C R d i m M., Let w : C N --* C n b e a linear projection to a complex subspace t h a t m a p s Tp(M) to C n isomorphically. T h e n t h e complex tangent spaces to N* ( M ) are p r o j e c t e d to C n injectively. By Corollary 3.2 applied to N * ( M ) , every lift A* of A is uniquely defined by t h e b o u n d a r y point A* (() and the w-component of t h e disc A, so the l e m m a follows. We consider the evaluation m a p s ~ - : Ap ~ M,
he(A) = A ( - 1 ) ;
G : Ap --* N p ( M ) ,
G(A) = [A~(1)],
where A~(1) = OA(1)/O~, is the derivative of A in the direction of the inner n o r m a l to bA at 1. By P r o p o s i t i o n 3.1, Ap is identified with a neighborhood of zero in the B a n a c h space of all analytic discsw in C n with w(1) = 0, where n = C R d i m M . T h e n ~" a n d G are s m o o t h maps, so we can consider their derivatives. We set
FA = a a n g e . T ' ( A ) C TA(-1)(M) GA = R a n g e g ' ( A ) C Np(M) M a i n L a m i n a 7.2. ([BRT], [T1]) Let A e Ap. Then the following two statements
hold. (F) FA = VA(--1) • C TA(-1)(M), in particular, d i m F A -- d i m M - d e f A and
T~(_D(M ) C FA. (G) GA = VA(1) • C Np(M), in particular, d i m G A = c o d i m M - d e f A . We first derive T h e o r e m 6.1 from Main L e m m a 7.2.
137
Proof of Theorem 6.1. We will show that there is a small disc A E Ap with d e f A -- 0.
Then by (G), GA -----Np(M). Therefore, F = C o n v (G(Av)) has nonempty interior, and Theorem 6.1 follows by Lemma 6.3. Let d = l i m i n f d e f A as A approaches the constant "disc" {p}. Let Ao E ,4p be a sufficiently small disc with defA0 = d. We can assume that A 0 ( - 1 ) = p, otherwise we replace Ao by the disc ~ H -40(() = A0((2), which has the same defect d and the property 4 0 ( - 1 ) = p. Since all discs A close to Ao have the same defect d, then by (F), the evaluation map ~ has constant rank dim M - d near Ao. Therefore, f maps a neighborhood of A0 to a smooth manifold S C M with dim S -- dim M - d. For every disc A close to A0, we have T ~ ( _ I ) ( M ) C FA = TA(-1)(S). Hence, S is a C R submanifold in M through p with CRdim S -- CRdim M. Since M is minimal, this can happen only if dim S = dim M, so defAo = d -- 0, what we need. The proof is complete.
8. P r o o f o f M a i n L e m m a We begin wth the following factorization lemma (see [BRT]). L e m m a 8.1. Let P be a smooth complex m x m matrix function on bA sufficiently close to the identity I and P(1) = I. Then P admits a decomposition of the f o r m P = R H so that R and H are smooth m • m matrix functions on bA close to I,
R(1) --- H(1) = I, R is real, and H holomorphically extends to A . Proof. The matrix H = R - 1 P is holomorphic and H(1) = I if and only if
(8.1) Let U -- R e ( R - 1 P )
Re(R-1P) = I -TIIm(R-1P). -- R - 1 R e P .
Since P is close to I, then R e P is invertible,
R -1 -- U ( R e P ) -1, and (8.1) turns into
(8.2)
U = I - TI(U(ReP)-IImP).
Since Im P is small, this equation has a unique solution U, which is real, close to I, and U(1) -- I. Then R :-- ( R e P ) U -1 is real, and (8.2) implies (8.1). Hence, H -- R - 1 p holomorphically extends to A. Lemma is proved. Let M be a generic manifold in C N defined by a local equation T ( z ) = o,
138
where r is a smooth R m valued function of Z = (z, w) E C m x C n = C N. We assume 0 C M and rz (0, 0) = I,
r~ (0, 0) = 0,
where rz = Or/Oz, r~ = Or/Ow. We denote by (w,~) the value o f t e T ; I ' ~
N) on ~ 9 Tp(CN). We then have
Tp(M) = {~ 9 Tp(cN) : Re (Or(p),() = 0}, Tp(M) = {~ 9 Tp(CN) : (Or(p),~) = 0}. In terms of the defining function r, the conformal bundle N*(M) is described by
Np(M) = {)~Or(p) : A 9 Rm}, where A is a row vector and r a column vector. Recall that Re (w, ~} defines the duality between N* (M) and N(M). Recall also that the operator of complex structure J gives an isomorphism T(M)/TC(M) ~ N(M) by [~] ~-* [J~]. Therefore Im (w, ~) defines duality between N*(M) and T(M)/TC(M).
Proof of (F) in Main Lemma. Since T(M) is orthogonal to N* (M) with respect to Re (., .), we mean by VA(--1) • the orthogonal complement with respect to (., .} or
Ira(., .). We first describe FA = Range~-'(A). The tangent space TA(AB) consists of all analytic dics A in T ( C N) attached to T(M) such that ~i(1) = 0 and for every ( 9 bA we have A(() 9 TA(r
that is Re
(Or(A(~)), A(r
= 0,
~ 9 bA.
Since ~ is an evaluation map, we have H ( A ) A = A ( - 1 ) and FA = { A ( - 1 ) : A 9 TA(.Ap)}. Since T(M) is orthogonal to N* (M) with respect to Re (., .), then for every A* 9
VA and every A 9 TA(Ap), we have R e ( A * , A ) = 0. The function (A*,A) is holomorphic in A, so it is constant.
Since A(1) = 0, we have ( A * , A )
-
O.
In
particular, ( A * ( - 1 ) , A ( - 1 ) ) -- 0. Thus FA C VA(--1) • To prove that FA D VA(--1) • T ~ ( _ I ) ( M ) such that r
we will show that for every real form r 9
= 0, there exists A* 9 VA such that r = I m A * ( - 1 ) .
We denote B(r
= rz(A(r
= (rz(A(r
r~(A(r
= (P(r
Q(r
r c bA.
139
We represent vectors by columns and covectors by rows. T h e n for A E TA(.Ap) we have Re (BA]bA) = 0. Likewise, A* E V A if and only if A*[bA = AB, where A is a •m-valued function such t h a t AB holomorphically extends to A. Let P = R H be the decomposition provided by L e m m a 8.1. T h e n by the substitutions
B ~ R-1B
(8.3)
(., 0i) 0
,
A ---* AR,
A ---* H A ,
we reduce the question to the case in which P - I , t h a t is B -- (I, Q). T h e n the condition Re ( B A ) = 0 for A = (s w) turns into
(8.4)
Re
+ Q(r162
= 0,
r
To satisfy this condition, we take an a r b i t r a r y analytic disc zb : A -~ C n with ~b(1) = 0 a n d define s
by
(8.5)
~ = - ( R e (Qzb) + iT1Re (Q(v)).
T h e real form r is defined for all Z = (z, w) E C N such t h a t Re ( B Z ) = Re (z + Q ( - 1 ) w ) = 0. T h e n r must have the form = Re (aw) - iA(z + Q ( - 1 ) w ) ,
r
where a E C n a n d A c R m are constant rows. T h e condition r
-- 0 takes
the form
(8.6)
Re (a~b(-1)) - iA(~(-1) + Q(-1)~b(-1)) = 0.
F i r s t restrict our a t t e n t i o n to A for which ~b(-1) ---- 0. As
T h e n by (8.6) we have
= 0, whence by (8.5)
(8.7)
(T1Re ( A Q w ) ) ( - 1 ) = O.
Note t h a t
(8.8)
2 f~
(Tlf)(-1) = ~
f(() d(
ebA ~ : - - 1 "
T h e condition (8.7) holds for all holomorphic zb with zb(=kl) = 0. T h e n by (8.8) the function AQ is orthogonal to all holomorphic functions in A therefore AQ extends to A holomorphically.
140
Now regardless of ~b(-1), we have TIRe (AQ~b) -- Im (AQ@), and (8.5) yields ABA = A(~ + Q@) - 0. In particular, ,~(~(-1) + Q ( - 1 ) ~ b ( - 1 ) ) = 0, and by (8.6) we have Re(c~@(-1)) -- 0. Since ~b(-1) is arbitrary, then a = 0 and r Im ( A B ( - 1 ) Z ) .
Since AB = (A, AQ) is holomorphic, then it represents A* E
--
VA.
Thus r ----I m A * ( - 1 ) , what we need. The proof is complete.
Proof of ((7) in Main Lemma.
The orthogonal complement in the statement of
N*(M)
(G) refers to the duality between
ip(U),
G'(A)ii = [iiv(1)] E
N(M)
given by Re (., .). We have
so
GA = a a n g e ~ ' ( A ) Recall that for every A* E
and
VA
= {[Av(1)] : A E
and every iil E
TA(.Ap), we
(A*(1),A~(1)) --
Differentiating this relation yields
TA(.Ap)}. have
(A*,A) --
0.
0 because A(1) = 0. Thus
GA C VA(1) • We now prove that
GAD
VA(1) •
that is G~ C VA(1). Let w E G~ C
N~(M).
The form w is represented by AB(1), where )~ E R m. The condition w E GA ~ means that for every A E
TA(Ap),
we have
Re(AB(1)A.(1))=
0. We need to show that
there exists A* EVA such that A*(1) = AB(1). By the substitution (8.3), we reduce the question to the case in which P - I, that is B -- ( I , Q ) .
(~, (v) E TA(.Ap), we
Note that B(1) --- ( I , Q ( 1 ) ) -- (I, 0). Then for every A = have
(8.9)
Re (A(~(1)) -- 0,
where ~ = c9~/0~. For every smooth harmonic function f in A with f(1) = 0 we have
(8.10)
Of(l) _ 1 ,/~o2~
f!eiO)dO_
i~
f(~)d~
We restrict to A for which @ = O(l~ - 112). Then (8.9) turns into Re
jo 2~ )~(e i~ dO
T~-I- F =0,
and by (8.4) and (8.10) we get Im ~ Eba
AQ(r162 de (( _ 1) 2 -- 0.
This relation implies that AQ is orthogonal to all holomorphic functions in A, hence AQ extends holomorphicMly to A. Now AB = (/k, AQ) defines A* EVA with A*(1) = AB(1), what we need. The proof is complete.
141
References
[AH]
Ayrapetian, R. A. and Henkin, G. M., Analytic continuation of CR functions through the "edge of the wedge", Dokl. Akad. Nauk SSSR 259 (1981), 777-781.
[BR]
Baouendi, M. S. and Rothschild, L., Cauchy-Riemann functions on manifolds of higher codimension in complex space, Invent. Math. 101 (1990), 45-56.
[BRT] Baouendi, M. S., Rothschild, L. and Tr6preau, J.-M., On the geometry of analytic discs attached to real manifolds, J. Diff. Geom. 39 (1994), 379-405. [BT]
Baouendi, M. S. and Treves, F., A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. 114 (1981), 387-421.
[Bi]
Bishop, E., Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1-21.
[Bo]
Boggess, A., CR manifolds and the tangential Cauchy-Riemann complex, CRC Press, 1991.
[Tr]
Tr6preau, J.-M., Sur le prolongement holomo~he des fonctions CR ddfinies sur une hypersurface reUe de classe C 2 dans C n, C. R. Acad. Sci. Paris S~r. I Math. 301 (1985, no. 3), 61-63.
[T1]
Tumanov, A. E., Extending CR functions on a manifold of finite type over a wedge, Mat. Sbornik 136 (1988), 129-140.
IT2]
__, On the propagation of extendibility of CR functions, In: V. Ancona et al.: Complex analysis and geometry. (Lect. Notes Pure Appl. Math. vol. 173) Basel, New York: Maxcel-Dekker, 1995.
DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS 1409 WEST GREEN STREET URBANA,ILLINOIS 61801 USA
C.I.M.E. S e s s i o n on " I n t e g r a l G e o m e t r y , R a d o n T r a n s f o r m s and C o m p l e x A n a l y s i s " List of Participants
A. ABOELAZ, Dept. of Math. and lnfo., Casablanca, Marocco F. ANDREANO, Via Cagliari 11, 00198 Roma, Italy F. ASTENGO, Dip.to di Mat. del Politecnico, Corso Duca degli Abruzzi 24, 10129 Torino, Italy L. ATANASI, Dip.to di Mat.,Univ., P.le A. Moro 5, 00185 Roma, Italy M.P. BERNARDI, Dip.to di Mat., Univ., Via Abbiategrasso 209, 27100 Pavia, Italy C. BOITI, Dip.to di Mat., Univ., Via Buonarroti 2, 56127 Pisa, Italy C. BONDIOLI, Dip.to di Mat., Univ. Via Abbiategrasso 209, 27100 Pavia, Italy L. BRANDOLINI, Dip.to di Mat., Univ., Via Saldini 50. 20133 Milano, Italy L. CAPOGNA, Dept. of Math., Purdue Univ., 1395 Mathematical sciences Bldg, West Lafayette, IN, USA L. CARBONE, Dept. of Math., Columbia Univ., New York, NY 10027, USA G. CARCANO, Dip.to di Metodi Quantitativi, Contrada S. Chiara 48/B, 25122 Brescia, Italy V. CASARINO, Dip.to di Mat., Univ., Corso Carlo Alberto 10, 10123 Torino, Italy P. CIA'I"rI, Dip.to di Mat. del Politecnico, Corso Duca degli Abruzzi 24. 10129 Torino, Italy S. COEN, Dip.to di Matematica, Univ., Piazza di Porta S. Donato 5, 40127 Bologna, Italy S. CUCCAGNA, Dept. of Math., Princeton Univ., 304 Fine Hall, Princeton, NJ 08544-1000, USA A. D'AGNOLO, Dip.to di Mat., Univ., Via Belzoni 7, 35131 Padova, Italy R. DAHER, Univ. Hassan II, Fac. des Sciences de Ain Chok, Casablanca, Maroeco C. DE FABRITIIS, SISSA, Via Beirut 2-4, 34013 Trieste, Italy G. DIN'I, Dip.to di Mat., Univ., Viale Morgagni 67/a, 50134 Firenze, Italy G. FELS, Univ. of Essen, Robert Kock Str. 1, Witten, Germany P. GRACZYK, D~pt. de Math., Univ. d'Angers, 2 bd. Lavoisier, 49045 Angers, Prance D. IACOVIELLO, Dip.to di lnfo. e Sist., Univ. "La Sapienza", 00185 Roma, Italy B. IVARSSON, Matematiska Institutionen, Box 480, 751 06 Uppsala, Sweden O. OKTEN, Dept. of Math., Stockholm Univ., 10691 Stockholm, Sweden M. V. OULD MOUSTAPHA, ICTP Maths Section, PO Box 586, 34100 Trieste, Italy C. PERELLI CIPPO, Dip.to di Mat. del Politecnico, Piazza L. da Vinci, 323, 20133 Milano, Italy A. PEROTYI, Dip.to di Mat., Univ., Via Saldini 50, 20133 Miolano, Italy M. PEVSNER, Inst. de Math., Univ. P. et M. Curie, 4 pl. Jussieu, 75252 Paris, Prance P. ROSSI, Dip.to di Mat., Univ. di Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy F. ROUVIERE, Univ. de Nice, Lab. Dieudonne, Parc Valrose, 06108 Nice Cedex 2, France D. ROUX, Dip.to di Mat., Univ., Via Saldini 50, 20133 Milano, Italy B. ROUBIN, Dept. of Math., Givat Ram,m The Ebrew Univ., 91904 Jerusalem, Israel E. SELIVANOVA, Rokossovskogo 17-113, Nizhni Novgorod 603136, Russia A. SELVAGGI PRIMICERIO, Dip.to di Mat., Univ., Viale Morgagni 67/a, 50134 Firenze, Italy T. KOBAYASHI, Graduate School of Math. Sciences, Univ. of Tokyo, Komaba, Meguro, Tokyo 153, Japan A. KORANYI, Math. Dept., Lehman College, Bronx 10468, USA L. LANZANI, Dept. of Math., Purdue Univ., 1395 Mathematical Sciences Bldg, West Lafayette, IN, USA H.-Q- LI, Dept. de Math., Univ. de Paris-Sud, 91405 Orsay, France R. MANFRIN, Scuola Normale Superiore, Piazza dei Cavalieri 7, 57126 Pisa, Italy C. MARASTONI, Dip.to di Mat. pura ed appl., Univ., Via Belzoni 7, 35131 Padova E. MIHAILESKU, Bd. Camil Ressu 27, Bloc NI, Sc.4m Ap183, Sector 3, Bucuresti, Romania C. MEDORI, SISSA, Via Beirut 2-4, 34013 Trieste, Italy F. MEYLAN, $bb Cigale, 1010 Lausanne, Switzerland C. MIRAN, Dept. of Math., Univ. of Ljubljana, Jadranska 19, 1 111 Ljubljana, Slovenia S. OFMAN, Inst. de Math., Grom. et Dynam., Univ. Paris 7, 2 place Jussieu, 75251 Paris, France L. SKRZYPCZAK, Fac. of Math. and Comp. Sci., A. Mickiewicz Univ., Matejki 48/49, 60769 Poznan, Poland M. SUNDARI, ICTP, PO Box 586, 34100 Trieste, Italy F. TONIN, Inst. de Math., Univ. P. et M. Curie, 4 pl. Jussieu, 75252 Paris, France S. TRAPANI, Dip.to di Mat., Univ. di Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy G. TRAVAGLINI, Dip.to di Mat., Univ., Via Saldini 50, 20133 Milano, Italy S. VENTURINL Dip.to di Mat., Univ., Piazza di Porta S. Donato 5, 40127 Bologna, Italy G. WEILL. Polytechnic Univ., 6 Metrotech Ctr. i, Brooklyn, NY 11201, USA A. ZAPPA. Dip.to di Mat., Univ., Via Dodecaneso 35, 16146 Genova, Italy
144
L I S T O F C.I.M.E.
1954
-
SEMINARS
1. Analisi
funzionale
2. Quadratura 3. Equazioni
1955
-
Publisher
4. Teorema
delle
C.I.M.E.
superficie
differenziali
di R i e m a n n - R o c h
non
e questioni
connesse
lineari
e questioni
connesse
5. Teoria dei numeri 6. Topologia 7. Teorie non linearizzate
in e l a s t i c i t i ,
idrodinamica,aerodinamica
8. Geometria p r o i e t t i v o - d i f f e r e n z i a l e
1956
-
9. Equazioni i0.
alle derivate
Propagazione
delle onde
ii. Teoria della funzioni funzioni
1957
-
-
aritmetica
13.
singolari
Integrali
e problemi
16. Problemi 17.
elettromagnetiche complesse
e delle
e algebrica e questioni (2 vol.)
attuali
di g e o m e t r i a
Ii principio
(2 vol.) connesse
in r e l a t i v i t &
differenziale
di m i n i m o
generale
in grande
e le sue a p p l i c a z i o n i
alle equazioni
funzionali
1959
- 18.
Induzione
e statistica
19. Teoria algebrica 20. Gruppi,
1960
- 21. 22.
1961
Sistemi
dei m e c c a n i s m i
anelli di Lie e t e o r i a
dinamici
e teoremi
Forme differenziali
- 23. Geometria 24.
e loro
del calcolo
delle
Teoria delle distribuzioni
25. Onde superficiali
1962
reali
automorfe
12. Geometria
15. V e d u t e
a caratteristiche
di p i ~ v a r i a b i l i
14. Teoria della turbolenza
1958
parziali
- 26. Topologia 27. Autovalori
differenziale e autosoluzioni
28. M a g n e t o f l u i d o d i n a m i c a
automatici della
(2 vol.)
coomologia
ergodici integrali
variazioni
(2 vol.)
145 1963
1964
- 29. Equazioni 30.
Funzioni
31.
Propriet~
- 32.
differenziali e variet~
di media e t e o r e m i
Relativit~
33. Dinamica
- 36. Non-linear
- 39.
di a n a l i s i
differenziali
continuum
non
numerica lineari
theories
Some aspects of ring t h e o r y
38. Mathematical
1966
in Fisica M a t e m a t i c a
generale
35. Equazioni
37.
di c o n f r o n t o
dei gas rarefatti
34. Alctule questioni
1965
astratte
complesse
Calculus
optimization
in e c o n o m i c s
of variations
Ed. Cremonese,
40. Economia matematica 41.
Classi
caratteristiche
42. Some aspects
1967
- 43. M o d e r n
questions
44. Numerical 45. G e o m e t r y
1968
- 46. 47.
1969
- 49.
analysis
of p a r t i a l
mechanics differential
bounded
equations
domains
and o b s e r v a b i l i t y
Pseudo-differential
operators
of mathematical
Potential
connesse
theory
of c e l e s t i a l
of h o m o g e n e o u s
Controllability
48. Aspects
e questioni
of d i f f u s i o n
logic
theory
50. Non-linear
continuum
theories
in m e c h a n i c s
a n d physics
and their applications 51. Questions
1970
of a l g e b r a i c
- 52. Relativistic
varieties
fluid d y n a m i c s
53. T h e o r y of group r e p r e s e n t a t i o n s
1971
54.
Functional
55.
Problems
- 56.
equations
and Fourier
analysis
and i n e q u a l i t i e s
in n o n - l i n e a r
analysis
Stereodynamics
57.
Constructive
58.
Categories
aspects
of f u n c t i o n a l
and c o m m u t a t i v e
algebra
analysis
(2 vol. )
Firenze
146 1972 - 59. Non-linear mechanics 60. Finite geometric structures and t h e i r applications 61. Geometric measure theory and m i n i m a l
1973
surfaces
- 62. Complex analysis 63. New variational techniques
in m a t h e m a t i c a l physics
64. Spectral analysis
1974 - 65. Stability problems 66. Singularities of analytic spaces 67. Eigenvalues of non linear p r o b l e m s
1975
~ 68. Theoretical computer sciences 69. Model theory and a p p l i c a t i o n s 70. Differential operators and m a n i f o l d s
Ed Li_uuori,
1976 - 71. Statistical Mechanics
Napoli
72. Hyperbolicity 73. Differential topology
1977 - 74. Materials with m e m o r y 75. Pseudodifferential operators w i t h a p p l i c a t i o n s 76. Algebraic surfaces
1978 - 77. Stochastic differential e q u a t i o n s 78. Dynamical systems
Ed Liguori,
1979 - 79. Recursion theory and c o m p u t a t i o n a l
Napoli and B i r h ~ u s e r V e r l a g
complexity
80. Mathematics of b i o l o g y
1980
-
81.
Wave
propagation
82. Harmonic analysis and group r e p r e s e n t a t i o n s 83. Matroid theory and its a p p l i c a t i o n s
1981 - 84. Kinetic Theories and the B o l t z m a n n E q u a t i o n
(LI~M 1048) S p r i n g e r - V e r l a g
85. Algebraic Threefolds
(I/qM 947)
86. Nonlinear Filtering and S t o c h a s t i c Control
(LNM
972)
(LNM
996)
1982 ~ 87. Invariant Theory 88. Thermodynamics and C o n s t i t u t i v e E q u a t i o n s
(LN Physics 228)
89. Fluid Dynamics
(LNM 1047)
147
1983
90.
-
Complete
Intersections
91. Bifurcation 92. Numerical
1984
- 93. Harmonic
Methods
Mappings
94. Schr6dinger 95. Buildings
1985
- 96. 97.
(LNM 1092)
T h e o r y and A p p l i c a t i o n s in Fluid D y n a m i c s
(LNM 1127)
and M i n i m a l
(I.,NM 1161)
Immersions
and the G e o m e t r y
of D i a g r a m s
and Analysis
Some Problems
in N o n l i n e a r
Diffusion
98. T h e o r y of Moduli
1986
-
99.
Inverse
i01.
1987
- 102.
- 104. 105.
1989
1990
- 108.
"
(LNM 1224)
"
(LNM 1225)
Economics
(LNM 1330)
Optimization
(LNM 1403)
Relativistic
Fluid D y n a m i c s
(LNM 1385)
in Calculus
Logic and Computer Global
- 106. Methods 107.
(LNM 1206)
Combinatorial
103. Topics
1988
(LNMlZ81)
(LNM 1337)
Problems
i00. Mathematical
Geometry
of V a r i a t i o n s
Geometric
Analysis
Topology:
109.
H
Control
ll0.
Mathematical
(LNM 1365)
Science
(LNM 1429)
and M a t h e m a t i c a l
of n o n c o n v e x
Microlocal
Physics
analysis
(LNM 1451)
(LNM 1446)
and A p p l i c a t i o n s
(LNM 1495)
Recent
(LNM 1504)
Developments
Theory
(LNM 1496)
Modelling
of I n d u s t r i a l
(LNM 1521)
Processes
1991
- IIi.
Topological
Methods
Differential
Algebraic
113.
to Chaos
Transition
(LNM 1537)
for O r d i n a r y
Equations
112. Arithmetic
Geometry in C l a s s i c a l
(LNM 1553) and
(LNM 1589)
Quantum Mechanics
i992
- 114. i15.
Dirichlet DLModules,
Forms
(LNM 1563)
Representation
Theory,
(LNM i565)
and Quantum Groups 116. N o n e q u i l i b r i u m Systems
"
(.T.,NM 1159)
Operators
Probability
Springer-Verlag
(LNM 1057)
Problems
in M a n y - P a r t i c l e
(LNM 1551)
148 1993 - 117. Integrable Systems and Quantum Groups
1994
-
(LNM 1620)
118. Algebraic Cycles and Hodge Theory
(LNM 1594)
119. Phase Transitions and Hysteresis
(LNM 1584)
120. Recent Mathematical Methods in
(LNM 1640)
Nonlinear Wave Propagation 121. Dynamical Systems
(LNM 1609)
122. Transcendental Methods in Algebraic
(LNM 1646)
Geometry
1995
-
123. Probabilistic Models for Nonlinear PDE's
(LNM 1627)
124. Viscosity Solutions and Applications
(LNM 1660)
125. Vector Bundles on Curves. New Directions
(LNM 1649)
1996 - 126. Integral Geometry, Radon Transforms
(LNM 1684)
and Complex Analysis 127. Calculus of Variations and Geometric
to appear
Evolution Problems 128. Financial Mathematics
1997 - 129. Mathematics Inspired by Biology 130. Advanced Numerical Approximation of
(LNM 1656)
to appear to appear
Nonlinear Hyperbolic Equations 131. Arithmetic Theory of Elliptic Curves
to appear
132. Quantum Cohomology
to appear
Springer-Verlag
F O N D A Z I O N E C.I.M.E. C E N T R O INTERNAZIONALE M A T E M A T I C O ESTIVO INTERNATIONAL M A T H E M A T I C A L S U M M E R CENTER
"Mathematics Inspired by Biology" is the subject of the First 1997 C.I.M.E. Session. The session, sponsored by the Consiglio Nazionale delle Ricerche (C.N.R), the Ministero dell'Universit~ e della Ricerca Scientifica 9 Tecnologica (IVI.U.R.S.T.),and European Community will take place, under the scientific directions of Professors VINCENZD CAPASSO (Universitb di Milano) and ODO DIEKMANN (University of Uu~cht) in Manina Franca (Taranto), from 13 to 20 June, 1997.
Courses
a)
Dyn=mlcs of physiologically structured populations (6 lectures in English) Prof. Odo DIEKMANN (University of Utrecht)
Outline:
I. Formulation and analysis of general linear models. The connection with multi-type branching processes. The definition of the basic reproduction ration R o and the Malthusian parameter r. Spectral analysis and asymptotic large time behaviou~. 2. Nonlinear models: density dependence through feedback via environmental interaction variables. Stability boundaries in parameter space. Numerical bifurcation methods. 3. Evolutionary considerations. Invasibility and ESS (Evolutionarily Stable Strategies). Relationship with fitness meal.ires.
4. Case studies:
cannibalism Daplmia feeding on algae - reproduction strategy of Salmon
-
References I.A.J. Melz & O. Diekmann (eds.),Dynamics of Physiologically Structured Populations,Lect.Notes in Biomath. 68, 19860 Springer. - O. Diekmann0 M. Gyllenberg0 1. A. J. Metz & H. R. Thieme, On the formulation and analysis of general deterministicstructuredpopulationmodels, L Linear theory,preprinL P. Jagers,The growth and stabilizationof populations, StatisticalScience 6 (I991 ),269-283. P. lagers,The deterministicevolutionof general branching populations, preprinl. O. Diekmann, M. A. Kirkilonis,B. Lisser, M. Louter-Nool, A. M. de Roos & B. P. Sommeijer, Numerical continuation of equilibriaof physiologicallystructuredpopulation models, preprint. S. D. Mylius & O. Diekmann, On evolutionarilystable lifehistories,optimization and the need to be specificabout density dependence, OIKOS 74 (1995),218-224. 1. A. J.Meetz, S. M. Mylius & O. Diekmann, W h e n does evolution opdmise? preprinL A. M. de Roos, A gende in~oduction to physiologicallystructuredpopulation models, preprinL F. van den Bosch, A. M. de Roos & W. Gabriel, Cannibalism as a lifeboat mechanism, 1. Math. Biol. 26 (1988), 619-633. V. Kaitala & W. W. Getz, Population dynamics and harvesting of semel-parous species with phenotypic and genotypic variabilityin reproductiveage, I. Math. Biol. 33 (1995), 521-556.
150 b)
When is space important in modelling biological systems? (6 lectures in n.,nglish). Prof. Rick DURRETI" (Comell University)
Outline: In these lectures I will give an introduction to stochastic spatial models (also called cellular automaLa or interacting particle systems) and relate their properties to those of the ordinary differential equations that result if one ignores space and assumes instead that all individuals interact equally. In brief one finds that if the ODE has an ~ n g fixed point then both approaches (spatial and non-spatial) agree that coexistence will occur. How if the ODE has two or more locally stable equilibria or periodic orbits? Then the two approaches can come to radically different predictions. We will illustrate these principles by the study of a number of different examples - competitionof Daphnia species in rock pools - allelopathy in E. coli spatial versions of Prisoner's dilemma in Maynard Smith's evolutionary games framework - a species competition model of Silvertown et al. and how it contrasts with a three species ODE system of May and Leonard -
References
-
c)
17,.DurreU (1955) Ten lectures on Particle Systems, pages 97-201 in Springer Lecture Notes in Math. 1608. R. Durrett and S. Levin (1994) Stochastic spatial models: a user's guide to ecological applications. Phil. Trans. Roy. Soc. B 343,329-350. R. Durrett and S. Levin (1994) The importance of being discrete and spatial. Theor. Pop. Biol. 46 (1994), 363-394. R. Durrett and S. Levin (1996) Allelopathy in spatially distributed populations. Preprint R. Durrett and C. Neuhauser (1996) Coexistence results for some competition models. Random walk systems modeling spread and interaction (6 lectures in English). Prof. K. P. HADh-:IFR (University of Tiibingen)
Outline: Random walk systems are semilinear systems of hyperbolic equations that describe spatial spread and interaction of species. From a modeling point of view they are similar to reaction diffusion equations but they do not show infinitely fast propagation. These systems comprise correlated random walks, certain types of Boltzmann equations, the Cattaneo system, and others. Mathematically, they are closely related to damped wave equations. The aim of the course is the derivation of such systems from biological modeling assumptions, exploration of the connections to other approachas to spatial spread (parabolic equations, stochastic processes), application to biological problems and presentation of a qualitative theory, as far as it is available 1. The problems and their history Reaction diffusion equations; velocity jump processes and Boltzmann type equations; correlated random walks; nonlinear interactions; Cattaneo problems; reaction telegraph equations; boundary conditions. 2. Linear theory Explicit representations; operator semigroup theory; positivity properties; spectral properties; compactness properties. 3. Semilinear systems Invarianee and monotonicity; Lyapunov functions; gradient systems; stationary points; global attractors. 4. Spatial spread Travelling fronts and pulses; spread of epidemics; bifurcation and patterns; Turing phenomenon; free boundary value problems. 5. Comparison of hyperbolic and parabolic problems.
References Goldstein, S, On diffusion by discontinuous movements and the telegraph equation. Quart. J. Mech. Appl. Math. 4 (1951), 129-156. Kac, M., A stochastic model related to the telegrapher's equation. (1956), reprinted Rocky Math. J. 4 (1974), 497509. Dunbar, S., A branching random evolution and a nonlinear hyperbolic equation. SIAM J. Appl. Math. 48 (1988), 1510-1526. Othmer, H. G., Dunbar, S., AIr, W., Models of dispersal in biological systems. J. Math. Biol. 26 (1988), 263-298. Hale, J. K., Asymptotic behavior of dissipative systems. Amer. Math. Soc., Providence, R.I. 1988. Temam, R., Infinite dimensional dynamical systems in mechanics and physics. Appl. Math. Series 68~ Springer 1988.
151 Hadeler, K. P., Reaction telegraph equations and random walks. Canad. Appl. Math. Quart. 2 (1994), 27-43. Hadeler, K. P., Reaction telegraph equations and random walk systems. 31 pp. In: S. van Suien, S. Verdoyn Ltmel (ods.), "Dynamical systems and their applications in science". Royal Academy of the Netherlands, North Holland 1966. d)
Mathematical Modelling in Morphogenesis (6 lectures in English). Prof. Philip MAINI (Oxford University)
Outline: A cenual issue in developmental biology is the formation of spatial and spatio temporal panems in the early embryon, a process k~own as morpboganesis. In 1952, Turing published a seminal paper in which he showed that a system of chemicals, reacting and diffusing, could spontaneously generate spatial patterns in chemical concentrations, and he termed this process "diffusion-driven instability". This paper has generated a great deal of research into more general reaction diffusion systems, consisting of nonlinear coupled parabolic partial differential equations (Turing's model was linear), both from the viewpoint of mathematical theory, and from the application to diverse patterning phenomena in development. In these lectures I shall discuss the analysis of these equations and show how this analysis carries over to other models, for example, mechanocbemical models. Specifically, I shall focus on linear analysis and nonlinear bifurcation analysis. This analysis will be extended to non-standard problems.
References
e)
G. C. Cruywagen, P.IC Maini, J. D. Murray, Sequential pattern formation in a model for skin morphogenesis, IMA J. Math. Appl. Meal. & Biol., 9 (1992), 22"/-248. R. Dillon, P.K. Maini, H. G. Othrner, Pattern formation in generalised Turing systems. I. Steady-slate patterns in systems with mixed boundary conditions, J. Math. Biol., 32 (1994), 345-393. P. K. Maloi, D. L. Benson, J. A. Shcrratr, Pattern formation in reaction diffusion models with spatially inhomogencous diffusion coefficients, IMA J. Math. Appl. Med. & Biol., ) (1992), 197-213. J. D. Murray, Mathematical biology, Springer Verlag, 1989. A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond., B 237 (1952), 37-72. The Dynlmlcs of Competition (6 lectures in English). Prof. Hal L. SMITH (Arizona Slate University, Tempe)
Outline: Competitive relations among populations of organisms are among the most studied by ecologists and there is a vast literature on mathematical modeling of competition. Quite recently, there have been a number of breakthroughs in the mathematical understanding of the dynamics of competitive systems, that is, of those systems of ordinary, delay, difference and partial differential equations arising in the modeling of competition. In these lectures I intend to describe some of the new results and consider their applications to microbial competition for nutrients in a chemostat.
References deMouoni, P. and Schiaffino, A., Competition systems with periodic coefficients: a geometric approach, J. Math. Biology, II (1982), 319-335. Hess, P. and Lazar, A. C., On an abstract competition model and applications, Nonlinear Analysis T.M.A., 16 (1991), 917-940. Hess, P., Periodic-parabolic boundary value problems and positivity, Longman Scientific and Technical, New York, 1991. Hirsch, M., Systems of differential equations that are competitive or cooperative. VI: A local C r closing lemma for 3-dimensional systems, Ergod. Th. Dynamical Sys., 11 (1991), 443-454. Smith, H. L., Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc. 1995. Smith, H. L., A discrete, size-sWactured model of microbial growth and competition in the chemostat, to appear, J. Math. Biology. Smith, H. L. and Waltman, P., The theo~' of the chemostat, dynamics of microbial competition, Cambridge Studies in Math. Biology, Cambridge University Press, 1995. Hsu, S.-B., Smith, H. L. and Waltman, P., Competitive exclusion and coexistence for competitive systems in ordered Banach spaces, to appear, Trans. Amer. Math. Soc. Hsu, S. B., Smith, H. L. and Waltmanm P., Dynamics of Competition in the onstirred chemostat, Canadian Applied Math. Quart., 2 (1994), 461-483.
FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER "Advanced Numerical Approximation of Nonlinear Hyperbolic Equations" is the subjectof the Second 1997 C.I.M.E. Session.
The session, sponsored by the Consiglio Nazionale delle Ricerehe (C.N.R), the Ministero dell'Universiti 9 della Ricerca Scientifica 9 Tecnologice (M.U.R.S.T.), and European Community wilt take place, under the scientific direction of Professor ALFIO QUARTERONI (Politecnico di Milano) at Grand Hotel San Michele, Cewaro (Cosenza), from 23 to 28 June, 1997.
Courses
a)
Discontinuous Galerkin Methods for Nonlinear Conservation Laws (5 lectures in English). Prof. Bemardo COCKBURN (University of Minnesota, Minneapolis)
Outline: Lecture 1: The nonlinear scalar conservation law as paradigm. The continuous dependence slruct'ure associated with the vanishing viscosity method for nonlinear scalar conservation laws. The enU'opy inequality and the enU'opy solution. The DG method as a vanishing viscosity method.
l..r~mre 2: Feam~s of the IX] method. The use of two-point monotone numerical fluxes as an artificial viscosity associated with intexr edges: Relation with monotone schemes. Slope limiting as an arl.ificial viscosity associate with the interior of the elements: Relation with the streamline-diffusion method. Lecture 3: Theoreticalpropertiesof the D G method. Stabilitypropertiesand a prioriand a posteriorierrorestimates. Lecture 4: Computational results for the scalar conservation law. Lecture 5: Extension to general multidimensional hyperbolic systems and computational results for the Euler equations of gas dynamics. b)
Adaptive methods for differential equations and application to compressible flow problems (5 lectures in English) Prof. Clacs JOHNSON (Chalmers University of Technology, Gtteborg)
Outline: We present a general methodology for adaptive error cona'o] for Galerldn methods for differential equations based on a posteriori error estimates involving the residual of the computed solution. The methodology is developed in the monograph Computational Differential Equations by Eriksson, Estep, Hansbo and Johnson, Cambridge University Press, 1996, and the companion volume Advanced Computational Differential Equations, by Eriksson, Esu~p, Hansbo, Johnson and Levenstarn (in preparation), and is realized in the software fem]ab available on Internet h t t : p : / / w ' , r ~ . m a t h . c h a l m e r s , s e / f e m l a b . The a posteriori error estimates involve stability factors which are
153 estimated through auxiliary computation solving dual linearized problems. The size of the stability factor determines if the the specific problem considered is computable in the sense that the error in some norm can be made sufficiently small with the available computational resources. We present applications to a variety of problems including heat conduction, compressible and incompressible fluid flow, reaction-diffusion problems, wave propagation, elasticity/plasticity, and systems of ordinary differential equations. Lecture 1: Introduction. A posteriori error estimates for Galerkin methods Lecture 2: Applications to diffusion problems Lecture 3: Applications to compressible and incompressible flow Lecture 4: Applications to reaction-diffusion problems Lecture 5: Applications to wave propagation, and elasticity/plasticity. l.,eeture 6: Applications to dynamical systems. c)
Essentially Non-Oscillatory (ENO) And Weighted, Essentially Non-Oscillatory (WENO) Schemes For Hyperbolic Conservation Laws (6 lectures in English) Prof. Chi-Wang SHU (Brown University, Providence)
Outline:
In this mini-course we will describe the constrdction,analysis, and application of ENO (EssentiallyNon-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws. ENO and WENO schemes are high order accurate finitedifference schemes designed for problems with piecewisc smooth solutions with discontinuities.The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the smoothest local stencil,hence avoiding crossing discontinuitiesin the interpolationas much as possible.The talkwill be basicallyself-contained,assuming only the background of hyperbolic conservation laws which will be provided by the fu'stlecturesof Professor Tadmor. Lecture I: ENO and WENO interpolation We ",viiidescribe the basic idea of ENO interpolation,sta,"dng from the Newton form of one dimensional polynomial interpolations.We will show how the local smoothness of a function can be effectivelyrepresented by its divided or undivided differences,and how this information can help in choosing the stencilsin ENO or the weights in W'ENO. Approximation resultswill be presented. Lecture 2: More on interpolation It -,rillbe a continuation of the fu'st.We will discuss some advanced topics in the ENO and WENO interpolation,such as differentbuilding blocks, multi dimensions including general triangulations,sub-callresolutions, etc. Lecture 3: Two formulations of schemes for conservation laws We will describe the conservative formulations for numerical schemes approximating a scalar,one dimensional conservation law. Both the cell averaged (finite volume) and point value (finitedifference) formulations will be provided, and theu" similarities,differences,and relativeadvantages will be discussed. ENO and WENO interpolation procedure developed in the firsttwo lectures will then be applied to both formulations. Total variation stable time discretization will be discussed. Lecture 4: Two dimensions and systems Is a continuation of the third. We will discuss the generalization of both formulations of the scheme to two and higher dimensions and to systems. Again the difference and relative advantages of both formulations will be discussed. For systems, the necessity of using local characteristic decompositions will be illustrated, together with some recent attempts to make this part of the algorithm, which accounts for most of the CPU time, cheaper. Lecture 5: Practical issues Practical issues of the ENO and WENO algorithms, such as implementation for workstations, vector and parallel supercomputers, how to treat various boundary conditions, cur-,'ilinear coordinates, how to use artificial compression to sharpen contact discontinuities, will be discussed. ENO schemes applied to Hamihon-Jacobi equations will also be discussed. Lecture 6: Applications to computational physics Application of ENO and WENO schemes to computational physics, including compressible Eulcr and NavierStokes equations, incompressible flow, and semiconductor device simulations, will be discussed.
154 d)
High Resolution Methods for the Approximate Solution of Nonlinear Conservation Laws and Related Equations (5 lectures in English) Prof. Eitan TADMOR (UCLA and Tel Aviv University.)
Outline: The following issues will be addressed. Conservation laws: Scalar conservation laws, One dimensional systems; Riemann's problem, Godunov, Lax-Friedrichs and Glimm schemes, Multidimensional systems Finite Difference Methods: TVD Schemes, three- and five-points stencil scheme, Upwind vs. central schemes, TVB approximations. Quasimonotone schemes; Time discretizations Godunov-Type Methods and schemes Finite Volume Schemes and error estimates Slreamline Diffusion Finite Element Schemes; The entropy variables Spectral Viscosity and Hyper Viscosity Approximations; Kinetic Approximations Lecture 1: Approximate solutions of nonlinear conservation laws - a general overview During the recent decade there was an enormous amount of activity related to the construction and analysis of modem algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present the successful achievements of this activity we discuss some of the analytical tools which are used in the development of the convergence theories associated with these achievements. In particular, we highlight the issues of compactness, compensated compactness, measure-valued solutions, averaging lemma,...while we motivate our overview of finite-difference approximations (e.g. TVD schemes), finite-volume approximations (e.g. convergence to measurevalued solutions on general grids), finite-element schemes (e.g., the streamline-dissusion method), spectral schemes (e.g. spectral viscosity method), and kinetic schemes (e.g., BGK-like and relaxation schemes). Lecture 2: Approximate solutions of nonlinear conservation law - nonoscinatory central schemes We discuss high-resolution approximations of hyperbolic conservation laws which are based on t centraldifferencing. The building block of such schemes is the use of staggeredgrids. The main advantage is simplicity, since no Riemann problems are involved. In particular, we avoid the time-consuming field-by-field decompositions required by (approximate) Riemann solvers of upwind difference schemes. Typically, staggering suffers from excessive numerical dissipation. Here, excessive dissipation is compensated by using modern, high-resolution, non-oscillatory reconstructions. We prove the non-oscillatory behavior of our central procedure in the scalar framework: For both the second- and thirdorder schemes we provide total-variation bounds, one-sided Lipschitz bounds (which in turn yield precise error estimates), as well as entropy and multidimensional L " stability estimates. Finally, we report on a variety of numerical experiments, including second -and third-order approximations of onedimensional problems (Euler and MHD equations), as well as two-dimensional systems (including compressible and incompressible equations). The numerical experiments demonstrate that these central schemes offer simple, robust, Riemann-sulver-free approximations for the solution of one and two-dimensional systems. At the same time, these central schemes achieve the same quality results as the high-resolution upwind schemes. Lecture 3: Approximate solutions of nonlinear conservation laws - convergence rate estimates Convergence analysis of approximate solutions to nonlinear conservation laws is often accomplished by BV or componsated-compacmess arguments, which lack convergence rate estimates. An L l -error estimate is available for monotone approximations. We present an alternative convergence rate analysis. As a stability condition we assume Lip + -stability in agreement with Oleinik's E-condition. We show that a family of approximate solutions, v ,ps which is Lip + -stable,satisfies it v ep,
(..t) - u ( . , t ) II t.,p -< C II v "p, ( . , 0 ) -
u(..O) ~1up
Consequently, familiar L ~ and new pointwise error estimates are derived. We demonstrate these estimates for viscous and kinetic approximations, finite-difference schemes, spectral methods, coupled systems with relaxation... Lecture 4: Approximate solutions of nonlinear conservation laws - the spectral viscosity method Numerical tests indicate that the convergence of spectral approximations to nonlinear conservation laws may - and in some cases we prove it must --- fail, with or without post-processing the numerical solution. This failure is related to the global nature of spectral methods. Since nonlinear conservation laws exhibit spontaneous shock discontinuities, the spectral approximation ponutes unstable Gibbs oscillations overall the computational domain, and the lack of entropy dissipation prevents convergence in these cases.
155 The Spectral Viscosity (SV) method attempts to stabilizethe spectral approximation by augmenting the latmr with high frequency viscosityregularization,which could bc efficientlyimplemented in the spectral,ratherthan the physical spare. The additionalSV is small enough to retainthe formal spectzalaccuracy of the underlying approximation; yet, the SV is shown to Ix:largeenough to enforce a sufficientamount of entropy dissipation,and hence, by compensated compactness arguments, to prevnm unslable spurious oscillations.Rec.cnt convergence results for the SV approximations of initialand initial-boundaryvalue problnms will be surveyed. Numerical experiments will bc presented to confLrm that by postprocessing the $V solution,one recovers the exact entropy solution within spectralaccuracy. Lecture 5: Approximate solutions of nonlinear conservation laws: entropy, kinetic formulations and regularizing effects We present a new formulation of multidimensional scalar conservation laws and certain 2x2 one-dimensiomd systems (including the isentropic equations), which includes both the equation and the entropy criterion. This formulation is a kinetic one involving an additional variable callecl velocity by analogy. We also give some applications of this formulation to new compactness and regularityresults for en~opy solutions based upon the velocity averaging lemmns. Finally,we show that this kinetic formulation is in fact valid and meaningful for more general classes of equations likeequations involing nonlinear second-order terms.
FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER
"Arithmetical Theory of Elliptic Curves" is the subject of the Third 1997 C.I.M.E. Session. The session, sponsored by the Consiglio Nazionale delle Ricerche (C.N.R), the European Community under the "Training and Mobilty of Researchers" Programme and the Ministero dell'Universith e della Ricerca Scientifica e Tecnologica (M.U.tLS.T.), will take place, under the scientific direction of Professor CARLO VIOLA (Universit~ di Pisa) at Grand Hotel San Michele, Cetraro (Cosenza), f r o m 12 t o 19 J u l y , 1997. Courses
a) I w a s a w a T h e o r y for E l l i p t i c C u r v e s W i t h o u t (6 lectures in English). Prof. John COATES (Cambridge University)
Complex Multiplication.
Outline: Let E be an elliptic curve over Q without complex multiplication, and let p be a prime number. A considerable amount is now known about the Iwasawa theory of E over the cyclotomic Z v extension of Q, and some of this material will be discussed in the courses by Greenberg and Rubin. On the other hand, very little is known about the Iwasawa theory of E over the field F~r which is obtained by adjoining all p-power division points on E to Q. If G~o denotes the Galois group of F ~ over Q, then Gor is an open subgroup of GL2 (Zv) by a theorem of Serre, since E is assumed not to have complex multiplication. The course will begin by discussing some of the basic properties of the non-abelian Iwasawa algebra A -- Z v [[Gcr 9 It will then give our present fragmentary knowledge of various basic I ~ s a w a modules over A, which arise when one studies the arithmetic of E over F c r Much of the course will be concerned with posing open questions which seem to merit further study.
Reyerenr.es The course will only assume basic results about elliptic curves and their Galois cohomology, most of which are contained in - J. Si]verman, "The Arithmetic of Elliptic Curves", GTM 106, (1986), Springer.
b) I w a s a w a T h e o r y for E l l i p t i c C u r v e s . (6 lectures in English) Prof. Ralph GREENBERG (University of Washington, Seattle)
Outline: This course will present some of the basic results and conjectures concerning the behavior of the Selmer group of an elliptic curve defined over a number field F in a tower of cyclotomic extensions of F. We will mostly discuss the case where the elliptic c~trve
157
has good ordinary reduction at the primes of F dividing a rational prime p and the cyclotomic extensions are obtained by adjoin p-power roots of unity of F. One of the ma]u results we will prove is Mazur's "Control Theorem". The proof will depend on a rather simple description of the Selmer group which also will allow us to study specific examples. We will also discuss the :'Main Conjecture" which was formulated by Mazur and its relationship to the Birch and Swinnerton-Dyer Conjecture. If time permits, we will discuss generalizations of this Main Conjecture.
ReJerences 1) J. S[Iverman~ The Arithmetic of Elliptic Curves, GTM 106, Springer. 2) S. Lang, Cyclotomic Fields I and II, GTM 121, Springer. 3) L. Washington, Introduction to Cyclotomic Fields, GTM 83, Springer. 4) J. P. Serre, Cohomologie galoisienne, LN.~vI 5, Springer. 5) J. Tare, Duality Theorems in Galois Cohomology over Number Fields~ Proceedings of the ICM, Stockholm, 1962, pp. 288-295. In reference 1, one should become familiar with the Tate modale of an elliptic curve and reduction modulo a prime p. In references 2 or 3 one can find an introduction to the structure of finitely generated modules over the Iwasawa algebra. It may be useful to consult references 4 and 5 to become familiar with Galois cohomolo~-. c) T w o - d i m e n s i o n a l r e p r e s e n t a t i o n s o f G a l ( Q / Q ) . (6 lectures in English) Prof. Kenneth A. RIBET (University of California, Berkeley)
Outline: The general theme of the course is that two-dimensional representations of Gal (-Q/Q) can be sho~n to have large images in appropriate circumstances. The lectures will touch on topics to be selected from the following list: 1. Recent work by Darmon and Mere! on modular curves associated to normalizers of non-split Cartan Subgroups of G L ( 2 , Z / p Z ) . A preprint by these authors proves, among other things, that the Fermat-like equations xp § yp _- 2z p has only the trivial nonzero solutions in integers x, y and z when p is an odd prime number! Technically, the main theorem of the paper is an analogue of a result pertaining to split Cartan subgroups of G L ( 2 , Z / p Z ) which appears in B. Mazur's 1978 paper "Rational i~ogenies o] prime degree". The connection with two-dimensional representations of Gal(-Q/Q) comes about because Merel has used the Darmon-Merel theorem to prove that the rood p representations of G a l ( Q / Q ) arising from non-CM Frey elliptic curves have large images for p > 2. [To read about this work prior to the conference, d o ~ l o a d Darmon-Merel paper (http://w~r~. math. m c g i l l , c e d d a r m o n / p u b / W i n d i n g / p a p e r , html) and follow the references given in that article] .2. Semi.stable ~epresentations. In his 1972 article on Galois properties of torsion points of elliptic curves, J.-P. Serre proved that the nmd p representation of Gal(Q/Q) associated to a semistable elliptic curve over Q is either reducible or surjective, provided that the prime p is at least 7. In his 1995 Bourbaki seminar on the work of Wiles and TaylorWiles, Serre extended his theorem to cover the primes p = 3,5. Using group-theoretic results of L. E. Dickson, one may replace Z / p Z by a finite etale Z / p Z algebra. [See ( f t p : / / m a t h . b e r k e l e y . e d u / p u b / P r e p r i n t s / K e n - R i b e t / s e m i s t a b l e . rex).]. 3. Adelic representations. A recent theorem of R. Coleman, B. Kaskel and K. Ribet concerns point P on the modular curve X0 (37) for which the class of the divisor (P) (Poo) has finite order. (Here Por is the "cusp at infinity" on X0(37).) The theorem states that the only such points are the two cusps on X0 (37). Our proof relies heavily on results of B. Kaskel, who calculated the image of the "adelic" representation Gal(Q/Q) GL (2, R) which is defined by the action of G a l ( Q / Q ) on the torsion points on the Jacobian of Xo (37). Here R is the ring of pairs (a, b) E Z x Z for which a and b have the same
158
parity. In my lectures, I would like to explain how Kaskel arrives at this results. [Although neither Kaskel's work nor the ColemA--Kaskel-Ribert is available at this time, one might look at 6 of Part I of the Lang-Trotter book on "~obenius distributions" (Lecture Notes in Math, 504) to get a feel for this kind of study]. 4. l-adic representation attached to m o d u l a r f o r m s . L e t f l, . . . , f p be newforms of weights k l , . . . , kp possibly of different levels and characters. Let E = E1 x . . . x F_~ be the product of the number fields generated by the coefficients of the different forms. For each prime , the l-adic representations attached to the different fi furnish a product representation p~ : G a l ( - Q / Q ) ~ G L (2, E | Ql). in principle, we know how to calculate the image of p "up to finite groups", i.e., the Lie algebras of pGa/(Q/Q). I hope to explain the answer - and how x~e know that it is correct. [One might consult my article in Lecture Notes in Math. 601 and my article in Math. Annalen 253 (1980)]. d) Elliptic curves w i t h c o m p l e x m u l t i p l i c a t i o n and t h e conjecture of Birch and Swirmerton-Dyer. (6 lecture in English) Prof. Carl RUBLN (Ohio State University) Outline:
The conjecture of Birch and Swinnerton-Dyer relates the arithmetic of an el]iptic curve with the behavior of its L-function. These lectures will give a survey of the state of our "knowledge of this conjecture in the case of elliptic curves with complex multiplication, where the results are strongest. Specific topics will include: 1) Elliptic curves with complex multiplication. 2) Descent (following Coates and Wiles). 3) Elliptic units. 4) Euler systems and ideal class groups. 5) Iwasawa theory and the "main conjecture". ~ e f erence$
- Shimura. G., Introduction to the arithmetic theory of automorphlc functions, Princeton: Princeton Univ. Press (1971), Chapter 5. - Silverman, J., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Math. 151, New York: Springer-Verlag (1994), Chapter II. - de Shaht, E., Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Math. 3, Orlando: Academic Press (1987). - Coates, J., Wiles, A., On the conjecture of Birch and S~innerton-Dyer, Invent. math. 39 (1977) 223-251. - Rubin, K., The main conjecture. Appendix to: Cyclotomic fields I and II, S. Lang, Graduate Texts in Math. 121, New York: Springer-Verlag (1990) 397419. Rubin, K., The 'main conjectures' of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991) 25-68. -
FONDAZIONE CTM.E. CENTRO INTERNAZIONALE M A T E M A T I C O ESTIVO INTERNATIONAL M A T H E M A T I C A L SUMMER CENTER
"Quantum Cohomology" is the subjectof the Fourth 1997 C.I.M.E. Session. The session,sponsored by the Consiglio Nazionale delle Ricercbe (C.N.R), the Ministoro dell'Universit,~t9 della Ricerca Scientifica9 Tecnologica (M.U.R.S.T.), and European Community will take place, under the scientificdirectionof Professors P A O L O D E B A R T O L O M E I S (Universiu3 di Firenzc), BORIS D U B R O V I N (SISSA, Trieste) and C E S A R E R E I N A (SISSA, Trieste)at Grand Hotel San Michele, Cen..aro(Coscnza), from June 30 to July 8,1997.
Courses
a)
Recent Developments in Non Pertm'bative Aspecet of String Theory (8 lectures in English). Prof. C~sar G O M ] ~ (Institutode Matemlticas y FfsicasFundamental, Madrid)
The aim of the course will be to review recent developments in non perturbative aspects of su'ing theory. The main topics we will cover are: i) Heterotic type 11 dual pairs and quantum feld theory limits. We will concentrate our study in Calabi Yau manifolds which are K3 fibrations or elliptic fibradons. We will derive the exact Seiberg Witten solutions from the local geometry of the Ca]aN Yau manifold. ii) Mirror symmetry for K3. Picard lattice, geometric mirror and T duality. We will consider some examples related with D brahe probes. iii)D iestamons associated with divisors of arithmetic genus one, and non perturbative superpotentials in 3 and 4 dimensional susy gauge theories.
References I) S. Kachru and C. Vafa, Nucl. Phys. B450, 1969, 69. 2) P. Aspinwall, K3 surfacesand stringduality,bepth 9611137. 3) E. Witten, Nucl. Phys. B474, 1966, 343.
b)
I. 2. 3) 4) 5) 6)
Gromov-Witten Invariants and Some Appficatlom to Symplectlc Topology, Enumerative Geometry, Finer Theory and Hmniltonian Systems (8 lecture in English) Prof. Gang TIAN 0d.I.T.,Cambridge)
In thisseriesof lectures,the following topicswill be covered: Introductionto symplectic manifolds and pseudo-holomorphic curves; Moduli space of holomorphic stablemaps; Gromov-Witten invariantsof general symplectic manifolds; Associadvity of quantum cohomology; Some applicationsto symplectic topology and enumcrative geomcWy; Relationto Flocr theoryand Harniltoniansystems.
The material should be enough for eight lectures.If extra time remains something about mirror symmetric for Ca]abi-Yau manifolds willbe intzoduced.
References I.
K. Behrend, Gromov-Witten invariantsin algebraic geometry, preptint,alg-georn/960101 I.
160
2. 3. 4.
14. 15.
K. Behrend and B. Fantechi,The inu'insicnormal cone, preprint,alg-geom/9601010. K. Fukaya and K. Ono, Arnold conjecture and Gromov-Wittcn invariants,prcprint, 1996. M, Kontsevich and Y. Martin, GW classes,Quantum cohomology and enumerative geometry, Comm. Math. Phys. 164, (1994), 525-562. J.l.,i and G. Tian, Vh'mal moduli cycle and Gromov-Witteo invariants of algebraic varieties, preprint, alggeom/9602007 J. Li and G. Tian, Virtual moduli cycle and Gromov-Witten invariants of general symplectic manifolds, preprim (alg-geom), August, 1996. D. McDuff and S. Salamon, J-holomorptuc curves and quantum cohomology, University Lec. Series, vol. 6, AMS. Y. Ruan, Topological sigma model and Donaldson-type invariants in Gromov theory, Duke Math. J. 83, (1996), 4t51-500. G. Liu and G. Tian, Arnold conjecture for general symplectic manifolds, prcprint, August, 1996. Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Diff. Geom. 42, no.2, (1995). B. Siebert, Gromov-Witten invariants for general sympleetic mamfolds, preprint, 1996. C. Taubes, The Seiberg-Wittcn and the Gromov invariants, J. of Amer. Math. Sot:., (1996). G. "lima, Quantum eohomology and its associativity, Proc. of Ist Current developments in Mathematics, Cambridge, 1995. F,. Witten, Topological sigma models, Comm. Math. Phys. 118, (1988). E. Wittcn, Two dimensional gravity and intersection theory on modu/i space, Survey in Diff. Geom., 1992.
c)
Elliptic Quantum Groups and Algebraic Bethe Ansatz (8 lectures in English)
5. 6. 7. 8. 9. 10. 11. 12. 13.
Prof. Alexander VARCHENKO (University of North Carolina at Chapell Hill) The Bethe ansatz is a method to construct common eigenvcctors of commuting families of operators (transfer matrices) occurring in two-dimensional models of statisticalmechanics. Faddcev and Takhtadzhan [6] reformulated the problem as a question of r~presentationtheory: commuting families of transfermatrices are associated to representations of era'rain algebras with quadratic relations (now called quantum groups). Eigenvectors arc constructed by properly acting with algebra elemeats on 'q'tighcstweight vectors", In this form, the Bethe ansatz is called the algebraic Bethe Whereas this construction has been very successful in rational and trigonometric intcgrable models, its extension to ellipticmodels has been problematic, although the Bethe ansatz, in the case of the eight-vertexmodel, is known since Baxter's work [I]. Recently, a definitionof ellipticquantum groups Ex.n (O) associated to any symple classicalLie algebra g was given by Fr [2]. It is related to a q-deformation of the Knizlmik-Z, amolodchikov-Bcrnard equation on toil.The representationtheory of E~.~ (sl~) was described in [3] and the algebraic Bethe ansatz for E~,~ (sl~) was intxoducedin [4]. The conslructionis very close to the construction done in the trigonometric case in [6].The main difference is that transfermatrices act on spaces of vector-valued functions rather than on finitedimensional vector spaces. Another class of problems in which the Bethe ansatz has been applied successfully is the class of C.alogeroMoscr-Sutherlaod quantum many body problems on the line with ellipticpotentials.In the case of two bodies, the Bethc ansatz goes actuallyback to Hermitc, who solved in this way the generalized I..am~ equations, cf. [7].These integrable SchrSdinger operators admit a q-deformation due to Rujsenaars [5]. In the lcctmv.s we shall discuss elliptic quantum groups, their connections with two-dimensional models of sutfi.Cdealmechanics, Kaiz.hnik-Zamolodchikov equations, and the Rujsananrs operators.
References [I]
R.J. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropie spin chain I, I1 and IlL Ann. Phys. (N. Y.) 76, 1-24, 25-47, 48-71 (1973). [2] G. F'clder, Conformal field theory and integrable systems associated to elliptic curves, Proceedings of the Intexnational Congress of Mathematicians, Ziirich 1994, p. 1247-1255, Birkh~iuser, 1994; Elliptic quantum groups, preprint hep-th/9412207, to appear in the Proce.edings of the IC'MP, Paris 1994. [31 G. Fr and A. Varehenko, On representations of the elliptic quantum group Et. q (sl l ), CMP, 1996. [4] G. Feldcr and A. Varchenko, Algebraic Bethe ansa~ the elliptic quantum group E t .n (el 2 ), Nuclear Physics B 480 (1996), 485-503. [5] $. ~. M. Ruijsenaars' C~mplete integrabi~ity ~f relativistic Calager~-M~ser ~.stems and ~1lipticfuncti~n /dem/t/es, Commun. Math. Phys. 110, 191-213 (1990). [6] L.A. Takhtadzhan and L. D. Faddccv, The quantum method of the inverse problem and the Heisenberg XF'Z chain, Russ. Math. Surv. 34/5, 11-68 (1979). [7] E.T. Whittaker and G. N. Watson, A course o f modern analysis, Cambridge University press, 1915.
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