Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1268 S.G. Krantz (Ed.)
Complex Analysis Seminar, University Park PA, March 10-14, 1986
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Editor Steven G. Krantz Department of Mathematics, Washington University St. Louis, Missouri 63130, USA
Mathematics Subject Classification (1980): 32 A 17, 32 A 10, 32 B 10, 32 H 15, 3 2 M 10, 3 2 F 0 5 , 3 2 F 15 ISBN 3-540-18094-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18094-X Springer-Verlag New York Berlin Heidelberg
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 other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
This volume represents the proceedings of an intensive week of complex analysis at Penn State which was held during the week of March I0, 1986. attended by about fifteen people with similar interests, attended every lecture.
The conference was
and every participant
The result was an enjoyable and rewarding exchange of ideas.
The lead article in this volume is a rather personal assessment of progress in Several Complex Variables
in the past fifteen years.
Subsequent articles in the
volume point to a number of new paths which we expect the subject to follow.
We
hope that the volume will be especially helpful to students and new members
in the
field~ as well as to people who are already established. We are grateful to the Department of Mathematics and the College of Science at the Pennsylvania State University for funding this conference.
Steven G. Krantz St. Louis, Missouri March,
1987
USA
CONFERENCE PARTICIPANTS
David E. Barrett
Princeton University
Eric Bedford
Indiana University
Jay Belanger
Princeton University
Steven R. Bell
Purdue University
John Bland
Tulane University
Joseph A. Cima
University of North Carolina
John P. D'Angelo
University of Illinois
John Erik Fornaess
Princeton University
K. T. Hahn
Pennsylvania State University
Steven G. Krantz
Pennsylvania State University
Donald Rung
Pennsylvania State University
Rita Saerens
Michigan State University
Berit Stens~nes
Rutgers University
TABLE
OF
CONTENTS
Steven G. Krantz, Recent Progress and Future Directions in Several Complex Variables
I
David E. Barrett, Boundary Singularities o f Biholomorphic Maps
24
S. Bell, Compactness of Families of Holomorphic Mappings up
29
to the Boundary
43
J. S. Bland, The Imbedding Problem for Open Complex Manifolds J. S. Bland, T. Duchamp and M. Kalka, A Characterization of CP n Automorphism Group
by its 60
Joseph A. Cima and Ted Suffridge, Proper Mappings Between Balls in Cn
66
John P. D'Angelo, Finite-Type Conditions for Real Hypersurfaces in C n
83
John P. D'Angelo, Iterated Commutators and Derivatives of the Levi Form
103
John Eric Fornaess and Nessim Siboy, Plurisubharmonic Functions on Ring Domains
111
Robert E. Greene and Steven G. Krantz, Characterizations of Certain Weakly Pseudoconvex Domains with Non-Compact Automorphism Groups
121
Rita Saerens, Interpolation Theory in cn: A Survey
158
Berit Stensones, Extendability of Holomorphic Functions
189
Rece.t
Progress
and
Future
O;rect[ons
Steven
1
doubt
important fifteen cOmment the
Louis,
is
the
at
this
from
Ramirez,
and
and
events
am
which
the
want
A
the
of
the to
lot
of
on
do
most
ten
or
is
to
here
a n d on w h i c h
has
been
written
[RAM]).
the
to
be
the
problem
well-known.
consequences
1970"s.
by
And
§-Neumann
seem
the
early
the
last
representations
and
H~rmander here
list
integral
theory
of
in
of
a from
familiarity,
[HENI]
concentrate
occurred
I
some
impinge.
(see of
produce
~hat
construction
estimates
rather
to
variables
I have
conference others
63130
Introduction
not.
which
the
L2
I would
0
complex
pervasiveness
weighted
Therefore
I
V-~r;abJes
University
Missouri
qualified
several
fallout
significance
key
in
areas with
given
the
Henkin,
and
anyone
years--certainly on a f e w
talks
about
that
results
Cornplel
of M a t h e m a t i c s
Washington
Section
5everml
G. K r a n t z
Department
St.
;n
of a f e w
These
are
the
following:
(i) (ii) (iii)
The
Kohn-Nirenberg
The worm
domain
Bounded
strictly
(Diederich (iv)
Points
of
It
is m y
view
finite
that
type
mapping
these
and subelliptic
about
several
variables
The the the
work
fundamental prior
had
1930"s,
and
detailed
and
completely
change
to
the
proof
and,
been the
analysis
(see,
for
that
early
of
understanding
problem
form
for
the
related
altered in
results
the
way
particular,
by d o z e n s
that
they
we
have
of
think
changed
of r e s e a r c h .
original
careful
complex
estimates
(1974)
discoveries,
have
functions
1972)
theorem
mathematicians,
1977)
exhaustion
1977)
(Kohn,
other
the c o u r s e
and Fornaess,
plurisubharmonic
problem
Fefferman's
(1973)
(Diederich
and Fornaess,
~-Neumann (v)
example
occurred
Fefferman's of
the
instance,
to of
the the
local
and
the
of
that
in
the
Likewise,
the
a
The case
1950"s
positive Kohn's
much goes
entailed
points.
pseudoconvex
problem
geometry
this:
work
theorem,
pseudoconvex
strongly
Levi
[BER]).
is s i m p l y
even
mapping
stronql ~"
reduced solution of
has
1970"s,
of
into very Levi
in
the
included
definite
solution
of
a
Levi
the
~-Neumann
Neumann
problem
boundary
Fefferman's to s a y
that
deepest What
strongly
Fefferman's [FOSI]
to date
conjecturing
that
not
which
that
a
strongly
beginning
equivalent
a
domains
can It
could
approximated
domains. See
That
I for
In
next
the
listed
above
pseudoconvexity, that
I
am
complex
But and
techniques problem,
the
is
past
the
corona on
of
reader
definitions
Section
bear
I
The
a
The
most P
in
important the
locally
point).
We
now
correct,
fact
be
nearly
is
we
that w e a k l y
are
at only
pseudoconvex
strongly
weakly by
pseudoconvex
pseudoconvex strongly
[DFI])
came
has
decade
or
and
complex I shall
to r e f e r
domains
pseudoconvex
as q u i t e
of
of
a shock.
so.
many
from
stressed
of
the
which
(v)
about
several
Princeton I
am
most
most
novel
ideas
While
the
inner
functions
other
programs
have
they
not
discuss
to [HO]
be
theory
with
-
the
variables, not
(i)
learned
should
the
ideas
items
have
It
of
yielded
or
that
are
also salient
and had to
them.
[KRI]
or
[RAN2]
for
basic
follows.
Example,
the W o r m
Domain,
Phenomena
fact
a domain
use we
primarily
the d i s c u s s i o n
elementary
to
what done.
derives
Kohn-Nirenberg
boundary
like
be
and Related
point
for
point
In
as
point
(this would
known
convex
portion
mind
and
on
to
only
problem,
is a d v i s e d
which
(see
collection
my
several
this p a p e r
that
I would
remains on
this
the
convex
bounded
discussing
which
to
effects
theme The
it
pseudoconvex
been
had
points.
outside
case
for
what
which
in
the
sections
variables--that
school.
the
one
recently
approximately
known
smoothly
no
As
details.
vehicle
and
been
believed
is not
concentrating
familiar,
strong
pseudoconvex
further a
strongly
long
from
three
as
had
weakly
generally
such
Section
weakly
1972
it h a s by
of the
1970"s,
a weakly
results.
exhausted
was
that
useful
gives
the
is safe
represents
points.
pseudoconvex
a
of
It
including work
on,
early
that
domains. be
the
even
token,
be
[FE3],
the
points.
analysis
balls.
built
is not
form
same
to
by
conjecture
to u n d e r s t a n d
the
until
in
of
pseudoconvexity.
equivalence,
fact,
a
it
pseudoconvex
the
such
in
that,
deeper
[FE2],
of s t r o n g
biholomorphic
consideration pseudoconvex
much
that
still
with
a
domains
[FEI],
is
years,
By
in
others
weakly
biholomorphically
least
work
about
were
a
pseudoconvex
and
on
strongly
striking
to local
realize
at
required
is v e r y
analogy
twenty
hinged
carefully
experts
is, up in
[FEI]
understanding
thought 1972,
in
of
Folland/Stein
[FOK])
conditions
work
approximations
(see
about •
is
a the
strongly
pseudoconvex
existence
of a local
holomorphic
separatinq
function <) =
satisfies
{z
n
~2p
j,k=l
~zj~ k
for
~
at
E cn
= p(z)
P.
Indeed,
if
< O}
(P)wj~ k ~ Clwl 2
~lw E ~n
then Lp(z)
~
n ~ j=l
~P (P)(zj azj
- Pj)
n
a2p
j,k=l
~zj~z k
+
(P)(zj
- Pj)(z k - Pk)
= O} =
{P}
satisfies N when a
cO
local
> 0
convex,
domain
~ it
: Iz - PI
is small.
biholomorphic
strongly then
{z
near
P
is clear
that has
that
, Lp(z)
Alternatively,
change
say
< cO
of in
once
coordinates the
defining
new
P
will
The step
in
the
construction the
be a local
existence of of
is
information
[KR2]).
Finally,
Were
algebraic the
pseudoconvex
aforementioned are
p*(w)
~
the p* ~ ,
P i
function. (see
separating
is
a
(gotten
the
functions in
are turn,
the
provide
~ problem very are
from
cohomology
functions
for
which,
In
machinery,
a suitable
separating
regularity
critical
[BER]).
functions
by s o l v i n g
functions
in fact
conjecture,
locally true
~(z)
where
that
is
(see
closely basic
for
considerations.
domains
domains,
and
the C a u c h y - F a n t a p p i ~
separating
peaking
renders
( W l , o . . , w n)
functions
problem
using
optimal
holomorphic
to h o l o m o r p h i c
Levi
Holomorphic
about
which
there
- P j)
separating
functions
fundamental.
P
p*(w)
separating
holomorphic
separating
important
function
the
formulas
qlobal
problem)
convex
of
integral
holomorphic
related
holomorphic
holomorphic
solution
existence
local
of
near
that
of
~p(w) = ~ g P * ( P * ) - ( w j awj
near
notices
coordinates
function
the p u l l b a c k
one
is a d e f i n i n g
then
that
smoothly
biholomorphically the
pullback
~ ~ @P*(P*).(wj ~wj
function
for
bounded
equivalent
weakly
to w e a k l y
of
- P j)
,
the c o n v e x
domain,
would
give
a weak
local
boundary.
holomorphic
This would
separating
mean
P £ ~ FI {z In
1973
Kohn
proving
and
that
pseudoconvex
2 = has
local
that
the
a
that
point
= O} C_ 82
of
the
.
optimistic
boundary
program
of
the
by
smooth,
The
lesson
Such
purposes
to be
is
they
a
one
domains
are
is that
.
locally
constructed domains at
and
Sibony
(see
which
show
be m a d e
by
locally
of
the m o s t
type
(see
tractable but
in
invariant
the
to the
notion
version
of
2)
points.
attention
C 2,
only
Section
pseudoconvex
restricts
even
Kohn/Nirenberg
the b a d p o i n t s ,
finite
if one
biholomorphically
we
biholomorphically
examples
as s t r o n g l y
even
is
vanishes
In p a r t i c u l a r
necessarily
called
domains,
h
h
side.
vanishes
as g o o d
if
then
0
stronger
cannot
are
0
Hakim/Sibony
from
form
=
of
not
0
pseudoconvex
Levi
pseudoconvex
pseudoconvexity
is
Indeed
h(O)
obtained
the
points
learned
weakly
Izll 8 + 1 5 1 Z 1 1 2 R e ( z l )6 < O} 7
and
near
boundaries
that
the
+
function.
0
domain
weakly
the
order.
for m a n y
that
convexity
too s i m p l e - m i n d e d .
The
Kohn-Nirenberg
investigate
and
[512],
Another the m o r e
drive If
pseudoconvex,
and basis
{2"
is c a l l e d It w a s
give
to r e d u c e strongly
2j
[HSI],
~ ~j+l
papers
Q
of w e a k l y
=
' each
of some
j
2j
, If
was
then
[BLI],
[BL2],
this w o r k .
2
domains
the p r o b l e m 2
each does
to
Silov
pseudoconvex
domains domain
A
of
peopl~
points,
is s a i d Qj not
is
of
to the
to h a v e strongly
have
a Stein
then
: Q"
the N e b e n h ~ l l e commonly
if
of
The
an o v e r v i e w
the s t u d y
number peak
pseudoconvex
basis
a
functions,
is a ( p s e u d o c o n v e x )
neiqhborhood
neighborhood
phenomena.
[BEF]
2
inspired
separating
related
[FOI],
tractable
Nebenh~lle. Stein
example
holomorphic
boundaries,
a
at e a c h
this
the
neighborhood
subsequently
are
is,
and
[SII],
in
a biholomorphism
Hakim/Sibony
is far
destroyed
of
domain
understand
finite
, hp(z)
IZlZ212
in e v e r y
pseudoconvex
now
simplest
2
convex
even with
sort: to
on
have
that w e a k l y
and
< cO
separating
Kohn-Nirenberg
to
[HSI],[SII])
We
: Re z 2 +
holomorphic
often
equivalent
convex
(0,0)
in a n e i g h b o r h o o d
infinitely
hp
domain
holomorphic see
[KON]
point
{ ( Z l , Z 2) E C 2
no
: Iz - PI
Nirenberg
the
function
that
~ 2
, Q"
of
Q
supposed,
is s t r o n g l y
pseudoconvex}
. if not f e r v e n t l y
hoped,
that e v e r y
smooth
pseudoconvex this was
domain
probably
has
a
a bit
T = has
a
very
there
is
better
large
no
In
{(Zl,Z2
For
)
Nebenh~lle.
substantive
any
event, [DFI]
pseudoconvex lesson
to
:
To
reason
subtle
I
the
the
the
IZll
be
basis.
Hartogs
< Iz21
sure,
why
In
retrospect,
triangle
< 1 }
~T
smoothly
that
from
is o n l y
bounded
But
of
the
at
Lipschitz,
domains
but
should
be
p
in
a
remarkable
Diederich
smoothly It
look
was
lot
particular
point this
not
time
and
bounded
a
difficult
different
from
are
more
they
discovery
to m e n t i o n period:
exhaustion
pseudoconvex
for
1977 a
Nebenh~lle. domains
this
plurisubharmonic
function
in
much
gave
a great
1970"s.
this
bounded
when
domainU:
outside--in
along
smoothly
defining
the
remiss
strictly
surprise
nworm
non-trivial
domains.
came
a
pseudoconvex
research
be
any
with
that
convex
to
news
given
quite
than
would
bounded
it
domain
than
impetus
was
exhibited
accept
inside
good
neighborhood
behaved.
Fornaess
the
Stein
optimistic.
~
and
an
~
the
the
functions domain
> 0
~
such
one
piece
discovery [DF2]. then
that
of
of the
In fact,
there p ~
-
is (-
a
p)~
satisfies
(i)
~
(ii)
p
( 0
on
~c
~
Cz E ~
(iii) (iv) For a
If
many
good
the
plurisubharmonic ,
p =
purposes,
the for
in and
One example
studying
very and
papers
is
complex
is
bounded the
the
~
;
;
,
all
a
c
c
< 0
< 0
that
the
killed.
holomorphic
;
such
plurisubharmonic
program
example
important the
~ = said
two in
set
Section
Let
C C ~
there
on
~
that
K ~ ~c
exhaustion
function
Kohn~irenberg It
mapping
has
example
proved
problem
is and
particularly
(see
for
instance
[DF4]).
substitute
These
on
0
: ~ ~ c}
then
Diederich-Fornaess
[BEL]
~
Q
K CC ~
substitute
useful
no
is s t r i c t l y
{z to
variety
several
the
2
lesson
results
tone
Points
: p(z) be V
of
was
learned
Diederich
complex
and
variables
of
Finite
Type
and
~ C2
have
smooth
that
type
m
E
of
for
the
< O}
decade
Z+
from
the
Fornaess
for
finite
such
that
of
hard
research
that
Subelliptic
if
Kohn-Nirenberg
is that
is
followed.
Estimates
boundary. there
there
calculations.
is a
A
point
P E
nonsingular
Ip(v)l
while
there
is no n o n s i n g u l a r
Ip(v')l
The
notion
of
finite
pseudoconvexity points
of
type
I
- PI m + 2
v E V
V"
, v"
is u n o r i e n t e d :
are
points.
,
variety
pseudoconcavity. strongly
such
it
distinguish
turns
pseudoconvex points
that
E V"
it c a n n o t Thus
Pseudoconvex
are
out
that
points
always
and
of
between the
only
strongly
odd
type.
In
domain
( ( Z l , Z 2) boundary The
points
of
notion
of
strongly
pseudoconvex
only
of
consists (see
: IZ112 + Iz212k
form
(ei0,0)
finite
type
points
domains.
For
if
points
points
where
the
points
where
only
of
that
result,
U
each
of
is
of
us
generic
type
Levi
foliated of
is
2k
in
the
the
one,
then In
has
idea
boundaries
is a r e l a t i v e l y
form
zero
dimensional
of
- I
quantify
vanishes.
Levi by one
U
,
type
exceeding
form
the
{ 1}
to
U ~ 8~ ~ C 2
finite
point
are
helps
are
only
[KRI])
a
the
pseudoconvex
containing
As
- PI m+l
complex
~ Clv"
type
and
pseudoconcave the
4 Clv
infinite
that
of
smooth
open
subset
U
consists
other
words,
U
rank.
It f o l l o w s
complex
manifolds.
type,
and
that
is
a
contradiction. Continuing
to
(equivalent) smoothly vector
restrict
definition
bounded
field
domain
attention
of
finite
in
C2
in a n e i g h b o r h o o d
L =
8p ( p )
Then
L
, L
near
P
.
span Their
(over span
has
~) no
the
P
~
C2 If
, P E @~ of
8z 2
to
type.
, we ~
, and
=
8p (p) 8z 2
_
8z 1 complex
component
define
~'0 =
spanlR{L' ~-}
f'l =
spanlR{~'0,[~'0,L],[I'0,i-]}
now
give
p(z)
another
< 0}
~ 0
, we
to
@~
is a
define
a
by
8p ( p )
8z 1
@
8z 2
tanqent in
the
space
complex
Z = Im~@P (P) @ + @P (P) @ ] Laz! 8z I az 2 8z~
However
{z:
normal
at
points
direction
~j We
call
=
span~{~j_l,[~j_l,L],[~j_l,L]}
P
a point
with
non-zero
such
an
finite
in K o h n ' s
type
non-singular fields, show
which
type
in
direction
the
paper
we
are
and
equivalent. in
[KOI]
have
varieties
that,
which
finite
m
if
~m-I Z
contains
while
~m
no
element
does
contain
element.
Implicit of
of
component
is the
given, the
The
one
other main
~2
, finite
type
a subelliptic
estimate
for
fact
in
in
terms
thrust
the
of
of
of
points the
that
terms
are
~-Neumann
of
contact
commutators
Kohn's
P
two definitions
order
paper
of
[KOI]
precisely
problem
was
those
of
the
of
vector to
near
form
~u|l 2 x< C [ l l ~ u l l 2 ÷ l l ~ * u l l 2] holds.
Here
IIul c
is
O-order m
u
a
is a test
tangential
Sobolev
and
(or
subsequent
function
Sobolev
L 2)
norm.
work
in
supported
norm
of
Kohn
[GR]
in a n e i g h b o r h o o d
order
¢
,
estimated
c
[KR2]
showed
and
and
in
of
II II
terms that
P
is
of
the
this
,
the type
estimate
is s h a r p . Since correct ,
and
for
1972,
also
type
to
(in
terms
to
functions the
Kohn's terms
that
of
the his
D'Angelo program In
those (see
right
for
of
important ideals
[K02]
~-Neumann
problem
on
forms
condition
would
Kohn
a series
also
be
dimensions of of
and
to
notion
of
one type
holomorphic clear
finite complex in was
support
that
it
was
condition,
in
estimates. gave
a
sufficient
existence in
any
of
deep
subelliptic
dimension.
necessary. a
in
equivalent
became
¢
conditions
contact
this
the
for
and
At
He
about
the
protracted
estimates conjectured same
study
time, of
the
initiated.
of
papers,
result
signalled
[DAN]
be
initiated
theory
framework.
in
the
algebro-geometric in
subelliptic
work
Catlin
soon
value
sufficient
of
points
it
determine
a definition
order to
to sharp
problem,
While
peak
for
which
of
fields.
forms,
and
of
for
and
formulated
definition
about
work the
~-Neumann
[BLI],[BL2],[HSI]),
condition
of
necessary
terms
this
deal
determine
CBG]
vector
thinking
to
the
in
proved
great
right
Graham
dimension) and
a
type",
the
and
commutators
helpful
been
estimates Bloom
any
of
has
"finite
determine
two.
hypersurfaces
for
of
subelliptic
exceeding
not
there
analogue
Meanwhile,
of
D'Angelo points
Catlin
that
developed of
he
built
finite had
on
from type.
found
D'Angelo's
first His
the
principles
right
ideas
an
semi-continuity
and
theoretical use
his
own
deep
insights
tool
for
of
D'Angelo
type
into
in
and
Cn
construction
the
problem
Catlin
there
of
of
plurisubharmonic
subelliptic
evolved
the
functions
estimates.
following
From
as a
the w o r k
definition
of
finite
:
Definition: Cn
the
attacking
Let
~
is h o l o m o r p h i c
=
{z
and
£ Cn
~(0)
: p(z)
= P
< O}
then
~(~)
and
P £ 8~
.
If
~': C --~
define
v(po%) v(~)
where
v(*)
T(P)
of
denotes
P
the
order
of
T(P)
We
say
that
P
Catlin then if
a
have
[CAT21
only
real
point
variables
in
collection
It
theory
the
type.
admissible
that
type
an
of
progress
vital
hat It
as
([KR3],[KR4]). developments
the role A
been is
made
Bergman,
careful on
the to
on
the
domains
of
finite
of type
of
Cn
finite
type that
singular type,
in
t 2 (see
metrics, these
work
should .
and More
invariant
of
, n > 2
and
strongly
of
[FO$1]).
Kobayashi
3
ought
of
integrals,
[STI],
Catlin's in
Section
homogeneous
understanding
large
complex
results.
case
clear
and
a
of
research
area
finite
understanding
domains
with
domains
of
final
complex
us
increasingly
Caratheodory,
several
understanding of
each
results
future
special
(see
that
These
these
on
Lusin
spaces
analytic
performed. of
of
new
analysis
becoming
in
theorems,
led
in
[DF3]
domain
~uch
constructions
analysis
theorems,
harmonic
made
direction
in
already
covering
tools
such
Fatou
> 0
P
problem
real
a bounded
M
be
near
Fornaess
with
provide
basic
harmonic
functions has
operators,
play
of
and
M
applications
important of
number
They
<
J-Neumann
given
principle)
dramatic
study
domains
metrics,
new
the
pseudoconvex the
domain
progress
the
in
is
exceeding
years.
which
least
development
[NSWI],[KR4]).
a
Define
T(P)
for
that,
is a
important
on
(at
maximal
other
recently,
can
view
pseudoconvex
the
.
Diederich
means
not
if
8fl P
type.
there
fifteen
particularly
detailed
The
integral
last
if
near
type
most
only
pseudoconvex This
finite
domains
some
that
finite
type.
the
the
is m y
be
of
boundary,
of
of
function
*
of
~(~)
if a n d
holds
bounded
has
tome
contains
a
finite
~
proved
is
analytic
of
represent
to
P
~ sup
type
estimate
that
is of
finite
has
if
proved
boundary with
is of
subelliptic
and
vanishing
to be
will issues
lead
to
TJne p a p e r s of
points
of
the
point
and
enable
a new
invariant results
of D ' A n g e l o finite of
view
metrics in
[KR4]), type
(iv)
kernels finite (see
the
(such
as
(such
as
the
[F02]
Section [FEI],
Theorem:
[GRS],
Let
results
3
The
Fefferman
C.
Fefferman
21
and
~n
Let
extends
to a d i f f e o m o r p h i s m
Fefferman's
proof
of
are
cumbersome
this w a s
domains.
mappings. domains were
by
(e.g.
the
represent
quite Lie
Even it
readily As
leads
the
Fefferman,
the
be
of
of
type
of
this
topics
the
program
safely
called
details
stressed the
and
of
very
of
about
these
a class
mappings
machinery
[HELl),
chapter of
given
calculations
powerful
proof
the
holomorphic
explicitly
statement
our
@-Neumann
holomorphic
Cartan--see
form
details for
Besides
about
of
While
Fefferman's the
asymptotic the
the
proved
involved
of
as
result
a classical
of
that
Then
relevant
such
mappings
[HUA])
the and
theorem.
theorems such
inequalities.
(see
pseudoconvex
map.
of
More
consideration of
Results
result:
strictly
boundary
here.
impact
calculation
in
poins
of f i n i t e
striking
analysis the
(and r e l a t e d
moreover
to n e w
instance
at
first
be
finite
canonical
"
detailed
first
though
accessible; an
of
~2
the
now
should
to
a
by p o l y n o m i a l
can
analysis.
~I
very
difficult
of
near
a biholomorphic
the
theory
what
difficult,
already
to
of
points
bounded,
certainly
explicit
described
often
one
of
of n o n - c a n o n i c a l
and Related
following
be
presented
problem
It w a s Prior
proceeded
of
be
of
problem),
--~ 2 2
to
solution
Theorem
be s m o o t h l y 21
geodesics
is a c o n s i d e r a t i o n
of
the
metric
purposes
the Levi
Mapping
involves
Bergman
near
behavior
for
kernels)
construction
the
ponts
[NSWI],[KR3],
points
expansions
kernel)
proved
~:
(see
as
near
of for
in C2).
22
in
behavior
(vi)
such
boundary
near
will
[CATI]
integral,
type
and Neumann
and
Henkin-Ramirez
domains
too
Szeg~,
[NSW3]),
(see
integrals,
spaces"
they
the b e h a v i o r
type
(iii)
asymptotic
commutators
that
(i)
area
finite
Hardy
(v)
for
Lusin
of
"real
the B e r g m a n ,
quadratic
discussion
of
I hope
finite
[KR4]),
points
of
of
the
[NSWI],
near
[FOS2]),
(see
[GRS],
In
[STI],
of
and
view
to c o n s i d e r
points
theory
a valuable
of
geometry.
researchers
the
theory
[FES],
type
kernels
(see
the
algebraic
function
functions
(see
provide point
from
boundary
(ii) g
type
holomorphic
in this v o l u m e both
of
near
t2),
finite
of
generation
Littl~wood-Paley of
type,
are the
they
in c o m p l e x extremely theorem
Fefferman's
is
results
insights. this
last
remark,
Klembeck
[KL]
used
Fefferman's
lO
asymptotic
formula
boundary
behavior
domain.
One
metric
result
about
curvature, strongly ball.
this
Greene
inspiration,
maps
it
most
argument
Fefferman's
result, in
results Burns,
obtain
important the
Fefferman among
It
himself
an
less
is
equivalence
known
(however
pseudoconvex to
nothing
known
boundary.
the
developed.
even
the
Clearly
of
there
of
in to
of
there
are
finite
a
while sinqle
to
many
For w e a k l y
results
type
C2
work
to
has be
about
essentially
invariants
important
very point.
biholomorphic
mappings, in
two
equivalent.
here, at
by
gave,
whether
differential 3
done
Fefferman
deciding
obstructions
domains.
been
biholomorphically
biholomorphic
is m u c h
has
results
program
Tanaka
[GK4],[BEDI],[BO~]).
biholomorphic
case
for
following
invariants
self-maps
equivalence
[GK3], though
boundary about
Even
are
"potential
of
the
papers
that
be c a l c u l a t e d .
those
used
these
procedure
global
[GK2],
can
the
know
Poincare's
invariants
Fefferman's
about
see
by
A
that
these
biholomorphic
biholomorphic
domains,
smoothness is
to
shows
Immediately
invariants
is
domains.
before
the
points
that
theorem
to
completed
In
the work
(biholomorphic)
needs
[CM]
[FE3].
boundary
only
one
boundary
effective
stressed
apply
the
only
is
biholomorphic
pseudoconvex
([BSW],[BS])
Lu
of F e f f e r m a n ' s
Fefferman's
Wells the
the
group
of
([FEI],[GK4])
how
of
domain.
of
anticipated
on
group
the
sectional
Kelmbeck's
to c a l c u l a t e
extent
[FE2],
things,
be
complete, Much
in
that
of
result
theorem:
using
invariants.
Moser
into
work
pseudoconvex
should
and
insights
of
to the b o u n d a r y
showed
some
Shnider,
the
but
true
a
analysis
how
strongly
series
and
they to
deepest
other
strongly
Chern
that
for
smoothly
with
[GK3]),
Poincare
curvature
automorphism
detailed
learn
principle,
considered
were
[TAN]. Perhaps
be
[GK2],
asymptotic
pseudoconvex
holomorphic
Wong's
consequence
of
power
extend
can
Bun
the b o u n d a r y
invariants
in
of
transitive
to
program
with
exist
in p r i n c i p l e
on
sectional
constant
to d o a m o r e
profound
a
invariants"
proof with
thus
depends
boundary
biholomorphisms
new
the
strongly
Coupled
of
([GKI],
and
invariants
Their
a
domain
able
differential formal
ball.
were
the
a
the
Krantz
vindicates
calculate
of
manifolds
expansion
Probably that
to on
holomorphic
yields
of a d o m a i n
kernel metric
the
that
and
Berglnan Bergman
that
complex
pseudoconvex
asymptotic
the the
is
approaches
Qi-Keng
as
for of
in not
done
the been
in
this
to c o n s i d e r
both
area. Fefferman's extending detailed shall
theorem
survey
of
of
this
also the
work
a few remarks
explicit
creation
has
simplifying
only make
The the
and
nature a
of
delicate
inspired
many
people
biholomorphic
mapping
theorem.
was
Bedford
in
given
about
by
A
[BED2],
very and
I
it here.
Fefferman's
asymptotic
calculus
singular
of
expansion integrals
required
(Boutet
de
11
Monvel
and
more
Sj~strand
natural
by
operators). to
Clearly
bypass
the
contexts)
to
which
hinges
hand,
Fefferman's
that p a r t
S.
on
of
be
in
used
biholomorphic a
new
another
way
behavior
formal Yang
quite
explored
gave nor
On
the
one
boundary
only
of
argument
gives
regularity
from
Surely
Bell
is
the
the
Bergman
R
for
that
be b o u n d e d
if
21
has Condition
R
Bell asymptotic
and
and have
of
the
the m a p p i n g
then of
C~(~) are
is
the
to
smooth
they
are
in
with
the
purely
Webster, neither
and
fully
boundary
nice
of are
ideas here.
a certain hand,
amount
of
Nirenberg's
that
extracts
the
three
not
be s u r p r i s e d
to by
role of
authors
extend
C®
made
a
if this
of
simplify [BELL]
to come
the m a p p i n g
biholomorphic
f(~)
C~(~)
and
Bell/Ligocka
following:
I ~K(z,()
If
from
and this
properties
maps.
Bell's
K(z,()
is
the
dV(~)
One
pseudoconvex
any biholomorphic
of B e l l ' s domains
mapping
of
21
theorems
says
and
one
of
to
~2
extends
them
the c l o s u r e s .
Bell/Ligocka
expansion
f i
which
in a s e n s e
such m a p p i n g s
contribution
study
~
then
kernel
in
are
the
that
other
program
central
the
to
argument
initiated
important
in
Bergman
been
gives
the
I would
2)
~2
to a d i f f e o m o r p h i s m
very
around
generality.
domain
from
and
that
most
P2: should
a way
ideas w e r e
of several
Together
pieces.
been
projection
kernel
the o t h e r
connection
has
of s h o w i n g
On
far-reaching
has
discovery
a
other
some
system ideas
smoothness
bootstrap
in g r e a t e r
single
(for
be
theorem
On
Nirenberg,
opinion
horocycles
below.
the
(Bergman's
is a m a r r i a g e
these
most
Bergman
Condition
from
The
in
has
coordinates,
interior).
reducing
the
his
apparent
in m y
of
that
While
their
regularity.
worked
theorem
[BEL].
program of
C l+e
the
value
finding
coordinate
mappings the
theory
reflection
Fefferman's
on
There
Yang's
argument
would
mapping
kernel.
proof
integral
to this c h a l l e n g e
discovered
to the p r o b l e m
boundary.
of a r g u m e n t
for
which
from
sort
rose
representative
apply
smoothness
complete
great
in
the
work
geodesics
interest
trivial.
profound
a method,
hand
has
Bergrnan
special
of b i h o l o m o r p i c
mappings
to the
[WE],
become
developed,
biholomorphic C l+c
Fefferman's
metric
people
a
Bercjnan
and really [NWY]
paper
create
mappings
boundary
is
Several
his
of
the
of
Fourier
biholomorphic
Bergman
there
part
of
insights.
to
rediscovery
the of
this
context
(which
of
of
and
the p r o o f .
us with
expansion
properties
it
in m a k i n g the
in s i m p l i f y i n g
proof
analysis in
into
goal
a
softer
Webster,
could
one
find
points
provided
succeeded it
asymptotic
and
difficult
[BMS]
fitting
for
have the
succeeded Bergman
in
kernel
eliminating and
the
both
the
differential
12
geometry The
(study
standard
of
geodesics)
from
(though
no m e a n s
method
Condition
R for a d o m a i n
~-Neumann
problem.
the
kernel
trick
and
for
mapping making
2
Kerzman
the
[KER]
8-Neumann
first
off
information
and
vice
Proposition
([BEL]):
Let
g E L2(22 ) •
Then
This
projection
mapping in
which
Kohn's the
formula
theorem. to
consider
program
of
8-Neumann
biholomorphic of f i n i t e It
arbitrary complex that
is
biholomorphisms
continue
CI,
counterexamples that
when
have
a
the
may
may be
even
the
domains
only
turn
out
trying
The but
or
predictable
the
If
conjecture
subelliptic local
[WE]
forattacking
and
domains is
to
the
the
impetus
estimates
now
know
, just
to for
that
is
true
it
is
as been
R.
a
and
also
conjectured
C2
however
boundary.
We
(after
all,
C2).
the
Nebenh~lle
On
[WE]
do in
not these
still
hope
mappings
will
many
is
his
in
which
biholomrphic
[BARI]
but
examples,
domains
for
in one
Barrett
conjecture,
are
like
to
is p s e u d o c o n v e x
it h a s even
C 2 then
have
extend
correct,
certainly
that
a
geometric
the o t h e r
hand,
problem:
then
the
partial
for
the p r o b l e m is
sharpened
biholomorphic
diffeomorphically it
will
subelliptic
of
conjecture
we
boundary;
have
R
we
thing.
a tool this
Jacc~
framework
added
Condition last
boundary
[FR]
Indeed
u = det
theorem
22
There
remains
will
by
the c l o s u r e s .
pseudoconvex
situation
still
of
smoothness
the
work,
and
this
not
the w r o n g
estimates.
property
too p o w e r f u l
a
its
these
Condition
It
mapping
satisfy
least of
between from
conceptual
of w h i c h
moments,
,
do
when
be
possibility bounded
this
work to
smoothly
CO
amount
of
the
of
is at
to p r o v e
examples
one
pseudoconvex.
boundary
constructions this
not
kernel
subelliptic
Catlin's
to
counterexample of
have
QI
domains
from
a new
mappings.
to
In o p t i m i s t i c
counterexample
the
a beautiful
to e x p l o i t
be b i h o l o m o r p h i c ,
to g e t t i n g
domains
bounded
for
formula:
produced
that
and
[FR],
Bergman
to a d i f f e o m o r p h i s m
bounded
verifying
connection
able
domains,
conjectured
smoothly a
two
introduced
domains
Thanks
of
the
the
biholomorphic
extends
been
dimension.
produced
key
ideas have
problem.
smoothly
all
the
noticed he
also
theorem.
for
= u . ( ( P 2 ( g ) ) o ~)
learning which
mapping
type,
has
is
Bell's
mapping
estimates
was
22
Pl(U.(go~))
the
Bell
projection
~: 21 ~
of
only method)
about
versa.
the f o l l o w i n g
proof the
of s u b e l l i p t i c
problem;
reading
of
the
is by w a y
properties use
, and
by
differential at hand.
softer
operator What may
qlobal
focus, of
to the c l o s u r e s .
clearly
estimates,
our
mapping
not
being §,
depend
on
a delicate
are
probably
be m o r e
suitable
arguments.
Bolstering
13
this
point
proves
of
view
that
obstruction, If ~
we
then
if
f
The
not
in
solution
the
Thus
usual
a different
down. which
any
continue little work
of
in
~-closed bounded
Kohn
form
domains used
with
to
~
such
the
§
soft
smooth
pseudoconvex
on
solution
Condition
but
there Kohn
domain
that
§u =
problem,
but
is n o
it
the
the
sort
of
is
not
projection
analogue
and
the
R,
rather
uses,
of
for
Bell's
program
breaks
global
argument
by
As
far
real
in the
analytic Prior
It
[BJT], fact
the
Baouendi, mapping
boundaries
to
their
case
to
work,
(however
well-known
know,
the to
encouraging
which
work
"all
the
who
methods
the
techniques
had
of
been
partial
ones
Monge-Ampere
very
see
very
people
now
will
is
that
other
I
of
Fourier-Bros-lagnolitzer
that
as
work
a biholomorphic
pseudoconvex
unknown
complex
recent
that
[MOW]).
of
be
the and
with
ones
note use
the
boundaries.
in
the
It m a y
is shows
strongly
completely
problems.
of
suggests
which
the
the
like
besides
that
domains
past
the
problem
version
projection,
development [Bd-T]
special
were
a
scheme
Webster
problems.
mapping
he any
in g e n e r a l .
beyond and
take
equations,
R:
pseudoconvex There
coefficients
Unfortunately,
Treves
especially
~-Neumann
in w h i c h
Condition
mind.
smoothly
is
the m e t r i c this
theorems,
transform,
a
Bergman
pseudoconvex
Moser
mapping
is a
Neumann
result
work
known
should
paper,
the
usual
analytically
was
encounter We
is
encouraging and
two
f
if
smooth
with
metric.
someday
Another
that
to
of
Nevertheless,
Jacobowitz, of
the
formula
may
[BAR2]
to
bounded
comes
u
Kohn's
R for
projection
smoothly
Kohn
closure
u
Barrett
metric.
while
Condition in
the
of
obstruction
to
of
show
is a
paper
is g l o b a l .
attention
to
on
there
local
[K03]
arguments
then
recent
no
it e x i s t s ,
paper
coefficients
the
is
restrict
the
global
is
there
nobody
that FBI)
working
are
has
of (or
on
differential
connected
equation,
time'.
with relevant
investigated
the to this
possibility. Finally in
this
let m e
section
holomorphic maps
are
broader
as
problem.
proper
tractable
For of
variety The
of
to
learned
that,
biholomorphic light a
in for
on
the
recent
lemma
Bell's
problems
techniques
as
Schwarz
have
the
brought
have
shed
(see
These
interpolation
has
that been
We
example,
the
mappings
[GKS]).
also
mappings.
problem
variants
mention
have
at
H®
on
for
maps. original
in
this
many
for
volume,
interesting
functions,
and
been
described
study
of
should
proper
purposes, the
study
biholomorphic is
boundary
revealed
have the
And
development the
article
turn
that
bear
the
proper of
discovery
biholomorphic as well
this
mapping
as
of and
[KRS],
connections
with
prove
in a
useful
contexts.
study
of
proper
mappings
has
given
new
vitality
to
this
branch
14
of
the
field
division study
of
survey
because,
problems
for
(see
holomorphic
[BED2]
instance,
source
of
Section
It
is
easy
fortified
what
there
recently things after
is
by to
about
will
be
bounded
we
are
now
that
Fefferman's
I not
running
look
become
out
of
Now
that
back
and
Elementary analytic
we
see
but
neat
to
of
kernel
Condition
mapping and
observation
R
problem,
points
of
the m a p p i n g
the
and
strong
cannot ball
in
1988
have
the to
were years
domain because expect. and
then,
the
great
impact
that
understand
insights
What
bring?
than
of
we
maturity,
unsolvable will
and
partial a
better
reveal?
a
connection
problem; to
richer
pseudoconvexity
new
gave
variables
dynamic:
What
~-Neumann
complex
is
Lewy
eventually
problem
domains
the
plateau
the
Kerzman
type.
it
first
to
subellipticity
finite
tha£ a
pseudoconvexity
of
and
and
very
several
the w o r m
begin
several
historical
of
led
of w e a k
Bergman
rather
consideration
operator.
we
is
information
primarily
for
not
As
that
reached
a continuing
understanding
now
next
can
domains
1974.
of and
way.
as w e r e
would
out
conference,
no d e t a i l e d
we
suggested
a
jaded.
have
differential
A
arbitrary
suggests
continuation
been
running
function,
what
in
has are
this
years wonder
It
in the
problem
they
fifteen
completed,
pseudoconvex
that
corona
more
steam,
had
realize
had
essay
we
Strongly
be,
last
perspective
as e a r t h s h a k i n g
for
and
Bedford's
we
that
and
theorem
this
our
exhaustion
would
the
at
the
problem
these
we
that
suspected. can
solve
mapping
better, hope
are
experienced to
feel
is u n l i k e l y
Again,
work
that
that
domains. It
of
lectures
1970
with
definition
this material.
years.
longer
In
connections the
Remarks
myself,
is
1970.
which
mapping
achievements
I no
on
excellent
the
to
[BB]).
progress
fifteen
to
think, in
(see
much
plurisubharmonic
were
biholomorphic
subject
I
it w a s
more
someone
the
of
listening
results
the
on
next
article,
led
Concluding
including
understood.
or
If
the
pseudoconvex
barely
there
for
true,
than
back
established
has
information
4
feeling
people,
this
is
weakly
only
a
After
writing
different
look
left
do.
What
to by
many
and
correspondences
is a g o o d
and,
it h a s
[DF4],[BEC])
this
between
the
in
led
connections
for
the
new
insights
~-Neumann will
turn
among
the
problem, a solution
is ever we
15
We
now
know
problem
a
exhibited there
a
full
no
necessary
and
~
of
on
which
estimates;
bounded
uniform
is b o u n d
to g i v e
§-Neumann
for
[$13]
domains
The
conditions
the
Sibony
pseudoconvex
estimates.
sufficient
problem
domains
uniform
smoothly
are
the
hand
satisfies
on
has which
determination
uniform
insight
of
estimates
into
the
for
nature
of
metrics
in
pseudoconvexity.
I
have
already
harmonic
Fefferman's Range
It
invariant
think
variables
be
further
finite
the
are
a
lot
experience,
not
the
been
a
pseudoconvex
functions,
the sharp
plurisubharmonic finite
boys
us,
the
most
and
so
giving talks
the
were out
memorable
that
for
by w h a t same
some
mapping
and
Skoda
the
these
several
complex
widely
more
spaced
but
there
knowledge
zero
and
that
went
on.
This
we
those was
heard
last
were
Nevanlinna
domain, work
bounded
on
points
attended
merely
year.
an
strongly
Chern-Moser
of
who
not
the had
for
the
the w o r m
was
There
there
sets
Bloom/Graham
think
that
and
formulas
theorem,
~ problem, the
attended
variables. time,
on
I
talks
all
complex
with
I ever
integral
functions,
on.
to of
several
accurate,
lot
complex
subject
for
exhaustion
us,
in
need
points
subject.
Fefferman's
estimates
a
several
Henkin
theory
of)
be
conference on
the
still
complex partial
problem,
shown
part
and
developments:
of
influenced
However, the
in
new
work
type,
profoundly the
of
(a
this
several
group
nothingness
may
hand,
Lie
have
white
the
exciting
conference
domains,
invariants,
at
to bear' on
conference
number
of
and
geometry,
discoveries
described
calculate
research.
~-Neumann
problem new
to
of
that
in m a p p i n g
between
and
the
description
tools
most
Williamstown
enormous
of
His
to b r i n g
Probably
once
on
differential
of
[HEN2],
domains,
frontiers
from
others,
problem of
both
Henkin
is g o i n g
analysis,
study
field
of and
connections
important
[CI]
more
new
of
mapping
to
work
types
the
clear,
[DF4]
difficult
new
the
entire
results'.
from
and
of
invariant also
of w h a t
harmonic
the
Cirka
isolated
1975
one
As
lead
"an
is
much
subjects
and
as
of
It
general
many
the
can
role
and
deep on
equations,
variables. variables
a
be
explored.
type,
now
is
that
disciplines
[FEI]
explain
should
and
differential
in
metrics
certainly
the
Diederich/Fornaess
metrics
problems.
to
questions.
work
[RANI],
invariant
I
alluded
analysis
were
a bunch For
many
of of
a revelation. of
everything is
a single
that
I heard
sentence
during
spoken
by
those Stefan
three
weeks,
Bercjnan.
In
16
the m i d d l e
of
one
"I
think
that
at
representative
time--in was
fact,
right,
were of
don't
[BTU],
and
and
[VLA].
[OKA]
great
reveals, in
changed
(to
all
pages
Hartogs
extension
contain
geometric
translation
old
for
the
and
in the
Krantz.
books
I
do
the
only
they
programs not
think
of
read
the the
The
papers in
wish
of
[~L],
there
works
of
has
or
the
old
must
be
some
paper
[lIAR]
lot
more
than
Bercjnan
C2
and
that
I
of
completely
1906 a
of
Fuks
discussion
collected
But
old
Thullen A.
writings
The
contains
I
B.
language
it
domains
forgotten? and
Malgrange
better).
language.
have
Behnke
textbook
[FOR],
to
we
of
CCAR],
language.
of
effect
Cartan
the
descriptions
he
representative
people,
techniques
the
painful
The
sense,
and
the
realized.
variables.
phenomenon.
modern
to g r e a t
publication
arcane
ought
Berc_~nan
different
Greene
long--surely
into
later.
Forsyth
recent
that
was
several
contains
appearances,
88
But
pay
he
[BOM],
is v e r y
in
about.
studied
complex
buried
is
the
it
several
ideas
Hartogs
As
look
talking
what
been
coordinates),
said
to at
powerful
latter
and
attention
and
has
up
much
didn't
years
Martin
(the
representative
workers
other who
stood
We
by
yet
Bergman
mappings
modification)
has
many
anyone
Bochner
Vladimirov
five
talks,
biholomorphic
knew
Ligocka,
utility
[FUKI],[FUK2],
Oka
learned
how
know
nobody
suitable
full
minute
studying
rediscovered
Bell
their
I wonder I
we
(after
Webster,
that
twenty
coordinates'.
were
used
the
people
almost
as
coordinates
of
you
(see
C3
[BERG])
which
could
of the
defy
understand
them. In
some
theory,
the
between
1945
complex
variables.
the
first
~
and
part of
features
the
little
on
Very
the
work
with
of
what
last
methods
than
been
basic
ideas
etc.--and reading
do
of
especially the
work
of
about
happens
to
be
I
many
of
they
involve
methods; I
the
points
ought
he
on
finite
been
It c o u l d
several
techniques
of the
the
good rely
calculation
and
I am
of
when
papers
worm
be
place
of
time
one
domain
or
the
reminded
of
the
most
argued
accomplished
sheaf
the
best
the
like took
the
the
type, at
roots
that
the
grappled,
to be.
has
think
papers
of
in w h i c h
more
that
instead,
read
or
with
basic
level,
that,
in
the
"elementary"
theory.
not
the
remembered
which laid.
sheaf
developed.
the
is
years,
I certainly have
from
1970
Levi
with
off
which
were
pseudoconvexity
fifteen
is
us
of
since
example E.
cut
techniques
much
subject
work
When
E.
little
powerhouse
geometry,
to
century,
powerhouse
Kohn-Nirenberg
of
K~hler
managed
the
experimentation.
old
and
1965,
of
foundations of
development
problem,
advocate What
I
do
abandoning advocate
the is
a
subject--pseudoconvexity, a
re-reading
Hartogs
and
Oka
of
the
doesn't
beautiful
machines
reinvestigation domains
old help
of
solve
that the
holomorphy,
literature. us
of
Even any
if
current
17
problems,
it
is b o u n d
to s u g g e s t
a
lot
of
new
ones--new,
at
least,
to
us.
The
author
Science
gratefully
Foundation.
acRnowledges
partial
support
from
the
National
~8
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[ BAR2]
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[DF3]
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[DF5]
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[FEll
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for
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the
~ - Neumann
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•
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20
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[FE3]
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[FOS2]
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[FOIl
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BOUNDARY
SINGULARITIES OF BIHOLOMORPHIC M A P S D a v i d E. B a r r e t t Dept. of M a t h e m a t i c s Princeton University Princeton, N J 08544
Let D 1 a n d D 2 be relatively c o m p a c t d o m a i n s with s m o o t h boundaries contained in the complex manifolds M 1 and M 2, respectively, and suppose that there is a biholomorphic m a p F f r o m D 1 to D 2. Let F denote the graph of F and let F denote the closure of F in DIxD2. The sin_~ular support of F (denoted sing supp F) will be defined as the set of all points (p,q) in ~ \ F such that F cannot be extended to a diffeornorphisrn from a neighborhood of p in D1 to a neighborhood of q in D2. Thus sing supp F is a closed subset of b D l x b D 2. If M 1 and M 2 are Stein and D 1 is a strictly pseudoconvex d o m a i n or, m o r e generally, a pseudoconvex d o m a i n of finite type then it is k n o w n that sing supp F m u s t be e m p t y ([Be], [BBC],[C]). The following question is stillopen. M u s t sing supp F = ~
whenever
M 1 a n d M 2 are
Stein?
It w o u l d be nice to resolve this question in particular for the important special case w h e r e M I = M 2 = [ ; n and D 1 and D 2 are w e a k l y pseudoconvex. W e shall see b y examples below that sing supp F need not be e m p t y w h e n M 1 and M 2 are not Stein. Nevertheless sing supp F m u s t satisfy certain restrictions, as w e see by the following result. Theorem ([BalD. If D 1 a n d D 2 are pseudoconvex, or if both d o m a i n s h a v e real ana]ytic boundary, then sing supp F has no isolated
points. ( O b s e r v e t h a t if (p,q) is a n i s o l a t e d p o i n t of s i n g s u p p F t h e n it is i m m e d i a t e t h a t F e x t e n d s to a h o m e o m o r o h i s m f r o m a n e i g h b o r h o o d of p in D1 o n t o a n e i g h b o r h o o d of q in D2.) T h e t e c h n i q u e u s e d in [ B a l l to p r o v e t h e a b o v e t h e o r e m c a n be extended s o m e w h a t to produce other examples of sets w h i c h are too small to contain c o m p o n e n t s of sing supp F. Note also that b y shrinking the domains D 1 and D 2 w e m a y see that an affirmative a n s w e r to Question i would imply in particular that no c o m p o n e n t of sing supp F is contained
25 in a product U I x U 2 , w h e r e U 1 and U 2 are Stein open subsets of M l and M 2, respectively. Given, then, that b o u n d a r y singularities of biholomorphic m a p s do occur a n d that they m u s t satisfy s o m e restrictions it seems desirable to a t t e m p t to search for the rudiments of a structure theory of such singularities. A necessary step in this direction is to study s o m e examples. In S e c t i o n 1 b e l o w w e e x h i b i t e x a m p l e s of b i h o l o m o r p h i c m a p s w i t h s i n g u l a r b o u n d a r y b e h a v i o r ; in p a r t i c u l a r w e g e n e r a l i z e the e x a m p l e s of [Ba2]. In S e c t i o n 2 w e pose s o m e q u e s t i o n s r a i s e d b y t h e s e e x a m p l e s .
.~i. E;~amples.
T h e s i m p l e s t e x a m p l e s of b i h o l o m o r p h i c m a p s w i t h s i n g u l a r b o u n d a r y b e h a v i o r a p p e a r to be t h o s e w h i c h c o m e f r o m d o m a i n s w h i c h a d m i t a c o m p l e x f a m i l y of a u t o m o r p h i s m s . Consider, for instance, the m a n i f o l d M=(E x ? 1 a n d t h e d o m a i n D= U x Ip1 w h e r e U is t h e u n i t disc. A n y function f on D induces an automorphism F of D b y t h e f o r m u l a F ( z , w ) - - ( z , w + f ( z ) ) , w h e r e w is t h e s t a n d a r d i n h o m o g e n e o u s c o o r d i n a t e on [pl. If f h a s s i n g u l a r b o u n d a r y b e h a v i o u r t h e n so does F. M o r e g e n e r a l l y , let M be a c o m p l e x m a n i f o l d w h i c h a d m i t s a $ - a c t i o n s u c h t h a t t h e m a p (5×M --* M, (t,z) ~ t . z is j o i n t l y h o l o m o r p h i c in t a n d z. Suppose that M contains a (E-invariant relatively c o m p a c t d o m a i n D with s m o o t h b o u n d a r y a n d suppose further that D admits a ~-invariant holomorphic function f with singular b o u n d a r y behavior. Let F be the a u t o m o r p h i s m of D given by F(z) = F(f(z).z) and let V be the variety { z E M : t.z=z for all t ]. Then b y linearizing the (E-action locally at points of M x V , w h e r e V is the variety { z E M : t.z=z for all t}, it is easy to see that sing supp Y contains at least the set { (z,t.z) :z
£ (sing s u p p f ) \ V ,
t ¢ cluster
set of f a t z }.
W e could of course replace (E b y a m o r e general complex Lie group in this set-up. To construct examples w h i c h do not fall within the f r a m e w o r k above w e begin with the following l e m m a .
L e m r n a . L e t k b e a p o s i t i v e i n t e g e r a n d let ~ d e n o t e t h e s e l f - m a p of t h e c o m p l e x p l a n e g i v e n b y ~(z)=z/z[ 2k. (~ is a h o m e o m o r p h i s m b u t n e i t h e r a b i h o l o m o r p h i s m n o r a d i f f e o m o r p h i s m . ) L e t f2 be a b o u n d e d
26 planar d o m a i n with s m o o t h boundary. Then the d o m a i n ~-If2 also has s m o o t h boundary. F u r t h e r m o r e if f2 has order of contact s %vith its tangent line at 0 then the corresponding order of contact for ~-ibf2 is s+2k(s-l). Proof. W e m a y a s s u m e that 0 £ b Q and it will suffice to restrict our attention to a neighborhood of 0. After a rotation w e m a y a s s u m e that b Q is 8;iven near 0 by the equation y+g(x)=0, w h e r e ~; is s m o o t h and vanishes to order s at O. T h e n b ~ - i Q is given near 0 b y the equation y+Izl-2kg(Izl2kx)=0. Using Taylor's t h e o r e m it is easy to check that the function y+Izl-2kg(Iz]2kx) is smooth and has non-vanishing gradient at 0, and furthermore that the first non-vanishing t e r m of its Taylor series w h i c h does not involve y is of order s+2k(s-l). [] In order to use the above l e m m a to construct biholomorphic m a p s w e first let M 2 be the quotient of ~×(E* by the properly discontinuous fixed point-free group of autornorphisms generated b y (z,w) ~
(o~z,o~w),
w h e r e cx is a real n u m b e r greater than i; w e continue to write points of M 2 in the form (z,w). Then the m a p ~: M 2 -* ~, (z,w) ~
lwl-lz
is a ~vell-defined submersion so that the relatively c o m p a c t d o m a i n D 2 = { - i ~ has s m o o t h boundary, w h e r e ~ is a n y d o m a i n satisfying the hypotheses of the L e m m a . ( W e shall a s s u m e that OEb/2.) N o w pick a positive integer k and let Iv[l be the manifold obtained from M 2 by dividing by the additional a u t o m o r p h i s m s (z,w) ~-~ (z,~'Jxv),j=O .....2k, w h e r e ~" is a primitive (2k+l) st root of unity. M 1 is biholomorphic to M 2 via the m a p F: M 1 -~ M 2, (z,w) ~ Let ~ be the submersion M I ~ compact domain
(z,z-2kw2k+l).
f:, (z,w) ~
lwl-lz. Then the relatively
27
D 1 =F-ID2 =F-l~-ig2= ~-l~-ig 2 has smooth b o u n d a r y b y the L e m m a . Thus F: D 1 -~ D 2 is a biholomorphic m a p of the sort w e wish to consider, and by looking at the images of rays of the form (pei@,w) with @ and w fixed and p decreasing to zero one can check that sing supp F = { ((zl,wl),(z2,w2)) : Zl=Z2=0 }; i.e.,sing supp F is a product of two elliptic curves.
Borrowing notation from the proof of the L e m m a w e note that D2 is defined near the points w h e r e z=0 by the equation r(z,w)=0, w h e r e r(z,w)=y+lwIg(]wl-lx) with z=x+iy. Recalling that the Levi form of bD 2 is given up to normalizations by
r
-dot
rz
r~ rz~ rgT r z ~
rw
rw~ rw~
it is e a s y t o c h e c k t h a t t h e o r d e r of v a n i s h i n g of t h e Levi f o r m of bD 2 a t p o i n t s w h e r e z = 0 is t h e n u m b e r s f r o m t h e L e m m a . Similarly, the c o r r e s p o n d i n g o r d e r of v a n i s h i n g for D 1 is s+2k(s-1). T h e g e o m e t r y of D 1 a n d D 2 a t p o i n t s w h e r e z = 0 is b e s t u n d e r s t o o d b y o b s e r v i n g t h a t a f t e r c h o o s i n g a b r a n c h of log z t h e m a p G: (z,w) (w-lz,z-2m/(l°g ~)) is a w e l l - d e f i n e d b i h o l o m o r p h i c m a p f r o m D 2 o n t o a c o m p o n e n t D 5 of t h e r e g i o n { (z,w) : exp(loglzI + i(2~)-11og c~-loglwl) cQ} in (t2; D 5 is of c o u r s e j u s t t h e R e i n h a r d t d o m a i n w h o s e l o g a r i t h m i c h u l l in IR2 is the planar d o m a i n obtained b y applying a branch of the complex logarithm function to the d o m a i n g2 and then stretching. The m a p G is well-behaved a w a y from z = 0. The examples in [Ba2] are obtained (after a change of coordinates) b y taking Q to be a disc.
2. Q u e s t i o n s .
L e t F be as in t h e i n t r o d u c t i o n . On t h e basis of t h e r a t h e r l i m i t e d s u p p l y of k n o w n e x a m p l e s w i t h sing s u p p F = ~ it is t e m p t i n g to ask if
28 t h e singularities of such a m a p m u s t propagate along varieties; m o r e precisely, w e pose the following question.
M u s t sine supp F be locally the union of complex analytic varieties of positive dimension ? Note that the m a x i m u m principle for strictly plurisubharmonic functions s h o w s that a n affirmative a n s w e r to Question 2 w o u l d imply an affirmative a n s w e r t o Question i. At this writing the author is unsure if the a n s w e r t o Question 2 is affirmative for all examples falling within the f r a m e w o r k of the opening paragraphs of Section i above. Note also that a u t o m o r p h i s m s play a role in both types of example considered in Section i above. (In the latter case w e h a v e an action of the two-torus ][2 on D 2 given b y (s,"c)-~(z,w)=(~"r/2nz,cxT/2nei@w) and a corresponding action of ~ 2 on D 1 such that F is ~2-equivariant.) The final question is offered as one w a y of asking w h e t h e r or not the appearance of a u t o m o r p h i s m s is in s o m e sense essential. Question 3. Can it happen that sine supp F is a closed complex manifold %vith discrete automorphism group ?
References.
[Ball [Ba2]
[15el [BBC] [C]
D. Barrett, Regularity of the B e r g m a n projection a n d local g e o m e t r y of domains, Duke Math. J° 53 (1986), 333-343. ---, Biholomorphic d o m a i n s with inequivalent boundaries, Invent. Math. 85 (1986), 373-377. S. Bell, Biholomorphic m a p p i n g s a n d the @-problem, Ann. Math. 114 (1981), 103-113. E. Bedford, S. Bell, a n d D. Catlin, B o u n d a r y behavior of proper holomorphic mappings, Mich. M a t h . J. 30 (1983), 107-111. D. Catlin, Subelliptic estimates for the [ - N e u m a n n problem on pseudoconvex domains (to appear).
Compactness of families of holomorphic mappings up to the boundarg
S. Belt Purdue University W. Lafayette, IN 47907
I. Introduction.
David Catlin has shown ([11,12]) that the Bergman projection
associated to a smooth bounded pseudoconvex domain of finite type (in the sense of D'Angelo [14]) satisfies strong pseudo-local estimates at each boundary point.
Thus,
Norberto Kerzman's proof [16} can be adapted to this class of domains and we are able to conclude that the Bergman kernel function associated to a smooth bounded pseudoconvex domain
O
of finite type extends
C~
smoothly to ~×~ minus the
boundary diagonal (see [6]). Recently, Harold Boas [9] and l [6] independently generalized Kerzman's theorem to a wider class of domains. In most applications of Kerzman's theorem to the problem of boundary behavior of holomorphic mappings, only the fact that the Bergman kernel extends smoothly to ~×£~ is needed. I intend to demonstrate in this paper that the full statement of Kerzman's theorem has important consequences in the study of families of biholomorphic and proper holomorphic mappings. Suppose that 01 and £~z are bounded pseudoconvex domains of finite type in Cn with
C~° smooth boundaries and suppose that {fi }
is a sequence of biholomorphic
mappings f i : 01---' 02" By passing to a subsequence, if necessary, we may assume that the
fi
converge uniformly on compact subsets of
f : O l - - ~ ~2-
~)1 to a holomorphic mapping
It is a classical theorem of Caftan [11] (see [17], page 78) which states
that f is either a biholomorphic mapping of 01 onto Oz or f is a mapping of 01 into b£~z, the boundary of 0 2, I Shall USe the full result on the smooth extendibility of the
30 Bergman kernel to prove
THEOREM i.
A)
In case f
is biholomorphic, the components of the mappings fi
converae in C°°(~ 1) to the corresponding components of f. B) I f f is a mapping of QI into
bO2, and if the inverse mappings Fi=f i -1
converge uniformly on compact
subsets of ~z to a mapping F, then there is a point Pl ~ bO~ and a point P2 E b~ 2 such that the mappings fi
converge uniformly on compact subsets of
~I-{P~}
to the
constant mapping f-=Pz.
It was observed by David Barrett [1] that Theorem 1, part A, follows as a consequence of the representation of the fi in Bergman-Ligocka coordinates used in [8] and the fact that pseudoconvex domains of finite type satisfy condition R. 1 will give a new proof of this result which will generalize to the case where f is a map into the boundary. Barrett's proof of Theorem 1, part A, and the proof given here are valid in the more general setting where O1 and Oz are smooth bounded domains in Cn which satisfy condition R. ( E. Bedford gave an alternate proof of this result in [3]. R. Greene and S. Krantz proved Theorem 1, part A, in [15] for strictly pseudoconve× domains.) Theorem 1, part B, sounds new even in the strictly pseudoconvex case. However, if Pl
or
P2
is a strictly pseudoconve× boundary point, then both domains must be
biholomorphic to the ball by Rosay's Theorem [181. Hence, the theorem is only interesting, and only new, in case Pl and Pz are weakly pseudoconvex boundary points. The hypothesis that the inverses converge in part B of Theorem 1 may seem strange. However, even in case O1 and 0 2 are both equal to the unit disc in C1, this hypothesis is necessary. Indeed, a typical sequence of automorphisms of the disc which converges to a boundary mapping is given by fk(z) = exp(iek)(Z-rkexP(i~ k))/(1 - zrkexp(-i~ k )) where {r k} is a sequence of real numbers 0
such that r k tends to one as k
tends to infinity. This sequence converges uniformly on compact subsets of the disc to
31
a boundary mapping if and only if
ek+~ k converges modulo 2~. The convergence
occurs up to the boundary away from a single point in the boundary if and only if we also have that 9k converges modulo 2:rr, and this happens if and only if the inverse mappings Fk=fk -1 converge uniformly on compact subsets of the disc. If the sequence {gk } forms a dense subset of the interval [0,2"n'] modulo 2"rr, then the mappings fk do not converge up to the boundary near any boundary point of the disc. It should be mentioned that, throughout this paper, the hypothesis that
0
be
smooth, bounded, pseudoconvex and of finite type can be replaced with the more general hypothesis that 0
be a smooth bounded domain whose Bergman projection satisfies
weak pseudo-local estimates (in the sense of [6]) at each boundary point. Alternatively, the finite type hypothesis can be replaced by the hypothesis that
0
is a smooth,
bounded pseudoconvex domain whose 8-Neumann operator is hypoelliptic. The methods used in the proof of Theorem 1 will also yield
Theorem 2.
If
fi : 01 ---' Oz is a sequence of biholomorphic mappings between
bounded weakly Dseudoconvex domains of finite type in boundaries which converges to a mapping f mappings
fi
Cn
with
into the boundary of
C°O smooth Oz, then, the
converge uniformly to a constant map on compact subsets of
01u F
where T denotes the set of strictly Dseudoconvex boundary Doints of 0~.
Note that there is no asssumption made about the convergence of the inverses in Theorem 2. Actually, the proof of Theorem I, part B, given below yields a stronger result than
the statement indicates. Let the Bergman kernel function associated to ~1 be denoted by Kl(z,w) and let ui = det [fi'] denote the holomorphic jacobian determinant of With a little additional effort, I can prove
fi
32 Theorem 3.
Lf f
is a maDDing into
bO2, and if the inverse mappings Fi= fi -1
converge, then, in addition to the conclusions of Theorem 1, the sequence of functions u i converges to the zero function in C°°(~F{pl}), f=P2 in C°°(~l-(5u{Pl}))
and the mappings fi convergeto
where 5 is the set of points z in
b 0 1 - {Pl} where
KI(Z,Pl)= O.
Note that the set S in Theorem 3 is rather small because, according to [4] (see also [8]), the function
Kl(z,p 1) cannot vanish identically in z on 01. Thus Kl(z,pl)
cannot vanish on any open subset of bOl. Tile set 5 need not be empty as Harold Boas' counterexample to the Lu Qi-Keng conjecture [10] reveals. With considerable additional effort, 1 can replace the set 5 in Theorem 3 by the smaller set consisting of points z in bO 1 where K1(z,p1) vanishes to infinite order in z.
I will only sketch the proof of this result in section 2. (This smaller set must
presumably be empty if 01 is of finite type, although I have not been able to prove this.) Versions of Theorems 1, 2 and 3 can be stated for proper holomorphic mappings. These theorems will be discussed in section 3 of this paper.
2. The p r o o f s of Theorems I, 2, and 3. Let
K1(z,w) and K2(z,w) denote the
Bergman kernel functions associated to ~i and Qz, respectively. Because ~i and ~2 are smoorh, bounded, pseudoconvex domains of finite type, Ki(z,w) C~(C~i×C~i -A i) where A i = { ( z , z ) : z ~bQ i) for
extends to be in
i : 1,2. Also, it has been shown by
Cattin ([12], [13]) that smooth, bounded, pseudoconve× domains of fintie type satisfy Condition R. Therefore, according to [4] (see also [8]), the complex linear span of the set A of functions h(z) such that h ( z ) : K2(z,w) for some w ~ D2 forms a dense subspace of A~(Q2). (Here, A~(~2) denotes the subspace of C~(D2) consisting of functions which are holomorphic on
D2-) The following fact is proved in [5] as a
consequence of the denseness of the span of A in A~(C22).
33 Fact I.
Given a positive integer s, there exist finitely many points wl, wz . . . . .
wM
in Oz with the following property. Given any function h in A°°(O 2) and any point p in ~2, there exist constants ci=ci(P) such that the function
M ~. i=l
r(z) :
agrees to order bounded by C
c i K2(z,w i)
s with
h(z)
at
p. Furthermore, the constants
ci
are uniformly
JJhlls where
Ilhlls : sup { I(aWaz
n(z) l : z E
I I -- s}
and C is a positive constant which does not depend on h or p.
1 will prove this result here for the convenience of the reader. Fix a point P0 in ~2- Let N be equal to the number of multi-indices ~x of length n and let
~xi, i=l,...,N,
differential operator
be an enumeration of these multi-indices.
Let
Di denote the
8~X/Sz°< with ~x = c~i. 1 claim that there exist points
in O2 such that the determinant of the matrix non-zero for
with 0 < t cxl <__s
z near P0- (Here, the operator
[eij] Di
wl,wz,...,w N
where eij(z) = Di Kz(z,w j)
is
is acting in the z-variable only.)
indeed, if det [eli(P0)] - 0 for all choices of (Wl,W2,...,wN) in (O2) N, then, because of the multi-linear property of the determinant and the denseness of the span of A°°(Oz), A°°(Oz).
it would f o l l o w that By letting
det[Digj(P0)] -- 0
for all functions
in
gl,g2,...,gN in
gi(z) = (z - p0 )~x with ~x = c~i, we obtain a contradiction.
constants c i = ci(p o) be given by the solution to the linear system
A
Let the
34
N Di h(p O) = ~ cj DiK2(Po,Wj) j=l
The constants properties at
PO
ci
and the points
The estimate
enlarging the set of points {w i}
w i, so chosen, clearly satisfy the required
I ci(P)l < C Ilhlls can be made uniform in p by if necessary because ~2
is compact and because
K2(Z,W) iS C°O smooth on ~2x£22. It is interesting to note that the points in the finite
set {w i} may be restricted to be in any open subset of 0 2 because, as was shown in [5], the linear span of functions of the form h(z) = K2(z,w) is dense in Aoo(O2) for
w
restricted to any open subset of £22. We will not need to know this more general result. Let Fi denote the inverse mappings to the fi and let Ui = det [Fi']. The Bergman kernels transform according to (2.1)
ui(z) K2(fi(z),w) = Kl(Z,Fi(w)) Ui(w) .
Let h(z) be any function in A°°(£22) and let s be a positive integer. Fact 1 together with the transformation formula for the Bergman kernels implies that there exist finitely many points {wj} in 02 with the following property. Given any point p E £21, there exist constants cj=cj(fi(p)) such that h(z) agrees to order s with the function H cj K2(z,w j) j=l
at fi(p). The constants cj are uniformly bounded independent of the choice of p. Now the transformation formula (2.1) implies that the function order s with the function
ui(z) h(fi(z))
agrees to
35 M ]"h(Z) =
~. j=]
cj Kl(Z,Fi(wj))Ui(w j)
(2.2)
at z=p. To prove Theorem 1, part A, note that the inverse mappings Fi converge uniformly on compact subsets of Qz to a biholomorphic mapping F : Qz f-1
' Q1. Of course F =
Thus, because the functions Fi and Ui converge uniformly on the compact set of
points {wj}, formula (2.2) reveals that the function ['h(Z) and its derivatives up to order s are uniformly bounded on QI, independent of the choice of p. Hence, we are able to conclude that the functions ui(z)h(fi(z))
together with their derivatives up to
order s are uniformly bounded on ~ql- Now, because s was arbitrary and because the functions ui(z)h(fi(z)) converge uniformly on compact subsets of Q1 to u(z)h(f(z)), we are able to deduce that ui(z)h(fi(z)) converges in A~(QI) to u(z)h(f(z)).
(Here
u(z) = det [f'].)
Let h(z)=l to see that ui converges in AC~(QI) to u. Since u=0 on D I, we may now let
h(z)
be equal to the j-th coordinate function
converges in A~°(Q1) to
(r)j.
zj
to deduce that
(fi)j
This finishes the new proof or Theorem 1, part A. Note
that we have not used the full hypotheses;
we only needed to know that Q1 and Q2
were smooth bounded pseudoconvex domains which satisfy condition R. We shall use a similar argument to prove Theorem 1, part B. First, note that the inverse mappings Fi also converge uniformly on compact subsets of Qz to a boundary mapping F : Q2 ---* bQlWe will now prove that there is a point P2 ( bQz such that fi(z)
tends to P2
uniformly on compact subsets of QI. To see this, note that ui(z) converges uniformly on compact subsets of
Q~ to the zero function as i tends to infinite according to
36
Cartan's theorem [11] (see also [17], page 78, Theorem 4). The transformation formula for the kernel functions can be written
ui(z) K2(ri(z),ri(w)) ui(w) = Kl(z,w).
(2.3)
Pick points z and w in O1 such that Kl(z,w)z O. Since ui(z) and ui(w) tend to zero, we deduce from formula (2.3) that Kz(fi(z),fi(w)) becomes large in modulus as i tends to infinity.
The smoothness of the Bergman kernel away from the boundary diagonal
now forces us to conclude that
fi(z)
and fi(w)
tend to the same point in
b£2z.
Suppose zl and z z are two points in O1. Let w E O~ be such that Kl(zl,w)z 0 and Kl(z2,w) ~ O. The above argument yields that fi(z~) and fi(zz) both converge to the same point in b£22; call it P2. (it should be remarked that this fact could have been deduced from the finite type condition on ~2 because a domain of finite type cannot contain a non-trivial holomorphic curve in its boundary [14]).
Similarly, it can be
shown that there is a point Pl E b£21 such that Fi(w) tends to p~ uniformly on compact subsets of £22. Let h(z) = 1. Arguing as in the proof of Theorem I, part A, we assert that given a positive integer
s, there exist finitely many points {wj}
in Oz with the following
property. Given any point z E Q1, there exist constants cj=cj(fi(z))
such that ui(z) =
ui(z)h(fi(z)) agrees to order s with the function M
rh(Z) :
• j=l
cj KI(z,Fi(wj))Ui(w j)
(2.4)
at z. The constants cj are uniformly bounded independent of the choice of z. Now, because s was arbitrary and because Fi(w j) ----' Pl and Ui(w j) ----, O, formula (2.4)
37
and the differentiability of Kz(Z,W) on ( 0 2 x O 2) - A 2 imply that ui tends to the zero function in C°°(~l-{pl}). We shall now use the transformation formula in the form (2.3) to prove that
fi
tends to the constant mapping P2 uniformly on any compact subset X of ~1 - {Pl}- Let z 0 be apoint in X and let w beapoint in g2~ such that Kl(z0,w);~ O. That sucha point
w
exists is an easy consequence of the denseness of the linear span of
{K~(z,w):wEOi} in A°°(E21)(see also [4]). Let V be an open neighborhood of z 0 in X such that there exists a positive constant c with ui(z)
tends to zero uniformly for
}Kl(z,w) I > c for all z E V. Since
z ~ V, and since
transformation formula (2.3) yields that
Kz(fi(z),fi(w))
ui(w)
tends to zero, the
becomes uniformly large in
modulus for z E V as i tends to inifinity. Thus, fi(z) becomes uniformly close to the boundary point P2 to which fi(w) converges. This finishes the proof of Theorem l, part B. The proof of Theorem 2 is accomplished by observing that, by Rosay's Theorem [18], if z is a strictly pseudoconvex boundary point of 0~, then for each compact subet K of Oz, there exists a ball B(z;6) about z in cn such that Fi(K) n B(z;6) = + for all i. Here, we have used the fact that because the domains are weakly pseudoconvex, they cannot be biholomorpnic to the ball. uniformly on compact subsets of
It will now be shown that
Ui tends to zero
Oz even if the sequence Fi does not converge.
Indeed, since IUi 12 is equal to the real jacobian determinant of Fi when viewed as a mapping from Rn into itself, it follows that the integral of
}Ui} 2 over a compact
subset K of 0 2 is equal to the volume of Fi(K). Since fi converg~sto a boundary mapping, the set Fi(K) is contained in a small neighborhood of hot for sufficiently large i. Thus, the volume of Fi(l() is small, and we deduce that U i(w) tends to zero uniformly on compact subsets of 0 2. It is now an easy matter to adapt the proof of
38
Theorem 1, part B, given above to the setting of Theorem 2. To prove Theorem 3, note that we have already shown that ui. tends to the zero function in C~°(~ 1 - {Pl}). To finish the proof, we must verify the following claim.
Claim !.
If zl and z 2 are two points in ~ - {Pl} and if Kl(zz,p 1) = O, then the ratio
ui(zl)/ui(z 2) tends to a finite limit as i tends to infinity.
To prove the claim, note that the transformation formula (2.0 implies that
UI(ZI)Kz(fi(Zl),W)
KI(ZI,Fi(W)) (2.5)
ui(z2)K2(fi(zz),W)
KI(Zz.Fi(w))
for any point w in O z. Choose w so that Kz(pz,w)= O. Then, since fi(zl) and fi(z2) tend to P2 and Fi(w) tends to p~, we obtain that the ratio ui(zl)/ui(z2) tends to the constant Kl(zl,pl)/Kl(zz,pl). The claim is proved. Pick a point
ui(z)/Ul(b).
b in 0 z such that
I now claim that
~(z) = Kl(z,pl)/Kz(pz,b). ~1-{Pl}
to
Kz(P2,b) ~" O. Define ~i(z)=
~'i converges in C~°(~ 1 -{Pl})
Indeed, that
to the function
~i converges uniformly on compact subsets of
~ follows imediately from the transformation formula in the form (2.1)
and the convergence properties of fi and Fi that we have already established. Let h(z) be any function in C°°(Ol-{pl})
A°°(Qz).
to ~(z)h(pz).
I will now prove that
~i(z)h(fi(z))
As before, let s be a positive integer and let {wj} be a
finite set of points in 0 2 with the property that given any point constants cj=cj(fi(z))
converges in
such that ~i(z)h(fi(z))
z ~ ~1, there exist
agrees to order s with the function
39
M r'h(Z) =
~ j=l
cj K~(z,Fi(wj))Ui(wj)/Ui(b)
(2.6)
at z. The constants cj are uniformly bounded independent of the choice of z. Claim 1 applied to the inverse mappings Fi now yields that the numbers Ui(wj)/Ui(b) tend to a finite limit as i tends to infinity. Thus we are able to conclude, by virtue of formula (2.6), that the functions
~i(z)h(fi(z))
together with their derivatives up to order
s
are uniformly bounded on compact subsets of ~1 - {P~}- Since ~i(z)h(fi(z)) converges uniformly on compact subsets of ~1 - {Pl} and because s was arbitrary, we conclude that the convergence must take place in C°°(~ 1 -{Pl}).
Note that the set
S in the
statement of Theorem 3 is precisely the set of points z
in bO~ - {p~} where ~(z) ~ 0.
Finally, by letting h(z) = zj, the j-th coordinate function, we deduce that the mappings fi converge in C°°(~ 1 - (5u{Pl})) to the constant map Pz. This finishes the proof of Theorem 3.
Remark.
To prove the stronger theorem mentioned after the statement of Theorem 3
in section 1, a division problem with uniform bounds must be solved.
We know that
~i(z)h(fi(z)) tends to ~(z)h(p z) in C*°(~l-{pl}) for each function h in A°°(O2) , in particular for h = z cx. The division theorem of [7] can be proved with uniform bounds at any point where ~(z) vanishes to finite order to yield that therefore fi(z) tends to Pz in C~ of the larger set.
3. The Proper Mapping Case. In this section, we will only consider families {fi } of proper holomorphic mappings fi : ~ ----' ~z between bounded pseudoconvex domains in Cn with real analytic boundaries. section will be given.
No proofs of any of the assertions made in this
40
If O1 and f2z are pseudoconvex domains with real analytic boundaries in Cn, then it follows from results in [2] that there are only finitely many possible proper holomorpnic mappings Pm: O1 ----' Oz, m=l,2,...,M, in the sense that given a proper holomorphic mapping f:Ol---* 02, there exists an automorphism { or 02 such that f= ~°Pm for some m, l<m<M. Hence, if
{fi} is a sequence of proper holomorphic
mappings between 01 and 02, then by passing to a subsequence, if necessary, we may suppose that there is a sequence of automorphisms
{{i} of 02 which converge to a
holomorphic mapping ~, : 0 1 - - ~ {')z and a proper holomorphic mapping P : O~----' O2 such that fi = {i ° P Thus, the study of the convergence properties of the sequence {fi } is reduced to Theorems 1, 2 and 3. We obtain
Theorem 4. I f
{fi } is a sequence of proper holomorphic mappings fi :O1--+O2
between pseudoconvex domains in Cn with real analytic boundaries which converges uniformly on compact subsets of O1 to a mapping f : 01 ----, Cn, then either A) there exist a proper holomorphic mapping P : O1----' Oz and a positive integer N such that for i > N, there is a sequence of automorphisms {{i } of £22 which converge i__n C°°(~ 2) to an automorphism { or oz with the property that fi = {i ° P for i > N. Thus, the sequence {fi } converges in C°°(O1) to the proper mapping {o P. O._E B) by Dassina to a subsequence, we may assume that there is a finite set of points {Pl,Pz,...,P k } c bO1 and a point q~bQz such that fi tends to q uniformly on compact subsets of ~1 - {Pl,,Pk}.
It is possible to prove versions of Theorem 4 in the finite type setting by applying the transformation formula for the Bergman kernel function under proper holomorphic mappings (see [5]) and arguing as in the proof of Theorems I, 2, and 3 above. The details of this argument will appear elsewhere.
41 F~f~r~nces I. D.E. Barrett, Regularity of the Bergman projection on domains with transverse symmetries, Math. Ann.258 (1982), 441-446. 2.
E. Bedford, Proper holomorphic mappings from domains with real analytic
boundary, Amer. J. Math. 106 (1984), 745-760. 3. E. Bedford, Action of the automorphisms of a smooth domain in Cn, Proc. A. M. 5. 93 (1985), 232-234. 4.
5. Bell, Non-vanishing of the Bergman kernel function at boundary points of
certain domains in C n, Math. Ann. 244 (1979), 69-74. 5.
6.
Bell,
Boundary
behavior
of
proper
holomorphic
mappings
between
non-pseudoconvex domains, Amer. J. Math. 106 (1984), 639-643. 6.
5. Bell, Differentiability of the Bergman kernel and pseudo-local estimates, Math.
Zeit.,in press. 7. 6. Bell and D. Catlin, Boundary regularity of proper holomorphic mappings, Duke Math. J. 49 (1982), 385-396. 8. 5. Bell and E. Ligocka, A simplicfication and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math. 57 (1980), 283-289. 9.
H. P. Boas, Extension of Kerzman's theorem on differentiability of the Bergman
kernel function, to appear. 10. H. P. Boas, Counterexample to the Lu Qi-Keng conjecture, to appear. 11. H. Cartan, Sur les functions de plusieurs variables complexes: L'iteration des transformations interieurs d'un domaine borne, Math. Zeit. 35 (1932), 760-773. 12. D. Catiin, Boundary invariants of pseudoconvex domains, Ann. Math. 120 (1984) 529-586 13. D. Catlin, Subelliptic estimates for the ~-Neumann problem on pseudoconvex domains, Ann. Math., to appear. 14. J. P. D'Angelo, Real hypersurfaces, orders of contact, and applications, Ann. Math. I 15 (1982), 615-637. 15. R. Greene and 6. Krantz, The automorphism groups of strongly pseudoconvex domains, Math. Ann. 261 (1982), 42B-446. 16. N. Kerzman, The Bergman kernel function. Oifferentiability at the boundary, Math.
42
Ann. 195 (1972), 149- 158. 17. R. Narasimhan, Several Complex Variables, Chicago Lectures in Mathematics, University of Chicago Press, 1971. 18. J. P. FIosay, Sur une caracterisation de la boule parmi les domaines de C n par son
groupe d'automorphismes, Ann. Inst. Fourier 29 (4) (1979), 91-97.
The Imbedding Problem For Open Complex Manifolds J.S. Bland
A problem of considerable interest in the study of several complex variables is to characterize which open complex manifolds are biholomorphically equivalent to open complex submanifolds of complex Euclidean space, and further, to characterize those which are equivalent to bounded open submanifolds. This problem is not only important in that it helps to delineate how bad higher dimensional complex manifolds can be (compare with the uniformization theorem of one complex variable), but it also has an intrinsic beauty in that it interrelates the various mathematical properties of the manifold. In the case that one is interested in proper imbeddings into C N (N > n, where n is the dimension of the manifold), it is well known that a necessary and sufficient cof~dition for the existence of such an imbedding is the existence of a smooth strongly pseudoconvex exhaustion function. More recently, this function theoretic property has been related to the existence of complete K~thler metrics with certain prescribed curvature properties (see section 1). In the case of bounded domains, it has long been recognized that various geometric properties of the boundary of the domain are related to the function theoretic properties of the domain. More recently, it has been recognized that this relationship can be extended to a certain class of open complex manifolds by associating to such manifolds a natural boundary, and that both the intrinsic function theoretic properties of the domain and the extrinsic geometric properties of the boundary can be related to the existence of complete Klthler metrics with certain additional prescribed properties. The main concern of this paper is to outline some of the results in these directions, and to indicate the various techniques needed to understand such manifolds.
§ 1: Stein Manifolds Stein manifolds form a broad yet cohesive class of open complex manifolds. The cohesiveness of the class is amply indicated by the two following well-known characterizations: An open complex manifold M n is Stein iff it admits a smooth strongly pseudoconvex exhaustion function iff it admits a smooth proper holomorphic imbedding into C 2n+1, [5, 25, 44, 45]. The breadth of the class is illustrated by the fact that in the uniformization theorem of one complex variable, both the unit disc and the complex plane are Stein. In this very fact, however, lies one of the major difficulties with
44 this class of manifolds:
it suppresses all information concerning the existence of
bounded holomorphic functions, and the vastly different function theoretic properties of the unit disc and the complex plane.
This section will concentrate upon geometric properties that guarantee that an open complex manifold is Stein, as well as more stringent geometric properties which will guarantee respectively the non existence of any bounded holomorphic functions and biholomorphic equivalence to C n. The remaining sections will study properties specific to bounded Stein submanifolds of C N (N >n).
In [26, 28, 29, 30], Greene and Wu determined several altemative formulations of sufficient differential geometric conditions to guarantee Steinness. They stated the following:
(1.1) Theorem. Let M be a complete K~hler manifold. Then M is Stein if any of the following holds:
(a)
M is simply connected with nonpositive sectional curvature
(b) M is noncompact with nonnegative sectional curvature, and moreover, the sectional curvature is strictly positive outside a compact subset of M. (c)
M
is noncompact with nonnegative sectional curvature, and the holomorphic
bisectional curvature is strictly positive (d)
M
is noncompact with strictly positive Ricci curvature, nonnegative sectional
curvature, and the canonical bundle is trivial. Notice that more can be said about the manifold in parts (b), (c) and (d). In each case, the manifold is Stein and cannot admit any bounded holomorphic functions; this latter fact follows from estimates by Yau [49] for harmonic functions on complete Riemannian manifolds with nonnegative Ricci curvature. This observation also indicates one strong motivation for working with K~thler metrics: on Kllhler manifolds, holomorphic functions are automatically harmonic.
The main technique in establishing this theorem is to use the differential geometric properties to establish the existence of a strictly plurisubharmonic exhaustion function. The various curvature assumptions on complete Riemannian manifolds are used to estimate the convexity of the distance function and functions derived from the distance function. Since the metric is K~hler, the geometric properties are closely tied
45 to the complex structure of the manifold; in particular, smooth strongly convex functions are strongly pseudoconvex (see [27]).
Siu and Yau used differential geometric properties in an elegant fashion in [47] to characterize manifolds which are biholomorphically equivalent to C n. They showed that if M is a complete open simply connected K~hler manifold with -A/12+e < sectional curvature < 0 for some positive constants A, ~ (r is distance from a fixed point p), then
M
is biholomorphic to C n. Their procedure to prove this theorem was
roughly as follows. They first solved the ~r problem on (n,1) forms in suitably chosen weighted L 2 spaces. By cleverly choosing the weights, they were able to establish the existence of a holomorphic n form which was nonzero at p
and satisfied certain
growth estimates at infinity; moreover they were able to show that any two holomorphic n forms satisfying these same growth properties were constant multiples of each other. They then changed the weight factors and solved another ~ problem on (n,1) forms to produce a holomorphic n form which vanished at p; the ratio of the new holomorphic n form to the first one was a holomorphic function vanishing at p and of "minimal" growth. This function acted like a linear holomorphic function on C n. They found n such functions forming a local coordinate system at p, and then showed that these functions indeed provided the components for a biholomorphic map from M to C n.
Mok, Siu and Yau weakened the geometric hypotheses needed to guarantee biholomorphic equivalence to C n. They established the following result [42]:
(1.2) T h e o r e m .
Let M n be an
n complex dimensional complete K~lhler manifold
(n >_ 2) with a pole p (i.e. - the exponential map centred at p is a diffeomorphism). Suppose that the norm of the sectional curvature is _< A/(1 + r2) l+e for some constants e, and A depending on e. Then M -- C n. If, in addition, M has nonpositive sectional curvature, then M is isometrically biholomorphic to C n.
Notice that this theorem not only gives intrinsic geometric characterizations of open complex manifolds which are biholomorphically equivalent to C n, but it also establishes an important gap theorem for complete K~hler manifolds with nonpositive sectional curvature: if the curvature dies too quickly at co, then the metric is identically flat. This latter phenomenon is really a differential geometric phenomenon; similar results hold for complete Riemannian manifolds [31].
46
§ 2: Geometry of Bounded Strongly Pseudoconvex Domains Bounded domains of holomorphy form an important subclass of all Stein manifolds. Their distinctive character arises from the fact that bounded domains admit lots of bounded holomorphic functions. (Compare, for example, the function theory of the unit disc with the function theory of the complex plane in one complex variable). Further, results of Hartogs, Levi and Oka lead one to consider the relationship between certain geometric properties of the boundary (pseudoconvexity) and the analysis on the domain. In particular, from knowing certain aspects of the function theory of the domain, one can deduce certain geometric properties of the boundary. Our approach to the study of bounded domains of holomorphy will be as follows. We will start with bounded strongly pseudoconvex domains, and observe that not merely the property of pseudoconvexity, but the entire CR structure of the boundary (that is - the restriction of the complex structure to the boundary) is invariant under biholomorphic maps. The CR structure on the boundary can be studied intrinsically, and in the analytic case, there exist a complete list of invariants. These invariants can in principle be determined from the asymptotics of complete metrics on the domain. observed that the boundary with its smooth CR
Finally, it will be
structure can be fully determined
from the asymtotics of complete K~ihler metrics on the interior: that is, open complex manifolds admitting complete Klthler metrics with certain asymptotic properties admit a natural compactification. The payoff for this approach will be in § 3 when we study the function theory of such open complex manifolds by alternatively considering them as the interior of compact complex manifolds with smooth boundary.
C. Fefferman established the following result in [21]: ff F : D 1 ~ D 2 is a biholomorphic map between two smooth bounded strongly pseudoeonvex domains in C n, then F extends smoothly to a diffeomorphism F : D1 -'~ D2- His proof involved two major steps.
The first step was a deep analysis of the Bergman kernel on smooth
bounded strongly pseudoconvex domains, and its singularity at the boundary. The second step used this information in an analysis of the geodesics for the associated complete Bergman metric; in this step, he showed that the geodesics, as curves in C n, extended smoothly to the boundary of the domain. Then, since biholomorphisms are isometries of the Bergman metric, the geodesics can be used to define biholomorphically invariant coordinate systems which are smooth to the boundary, and the map F must also be smooth to the boundary.
47 An immediate consequence of Fefferman's result is the following: If M is an open complex manifold which is biholomorphically equivalent to a smooth bounded strongly pseudoconvex domain D c C n, then there is naturally associated to M smooth invariantly defined boundary, and the complex structure on
a
M extends
smoothly to a strongly pseudoconvex CR structure on the boundary. Indeed, for a fixed biholomorphism M ~-D c c C n, define OM by M k..)OM--D.
That this
definition of the boundary is invariant follows from the fact that if M -- D' for any other smooth bounded strongly pseudoconvex domain D', then D - - M - - D ' ,
and by
Fefferman's result D --- D', where the last equivalence is a biholomorphism on the interior and a diffeomorphism on the closure.
At about the same time as Fefferman's results, Chern and Moser [17] studied the boundaries of smooth strongly pseudoconvex domains and were able to describe a complete list of local invariants. Moser's approach was extrinsic, describing a normal form for the boundary. Chern's approach was intrinsic, and generalizes immediately to abstract nondegenerate CR manifolds.
Let N be a nondegenerate CR manifold and 0 a nonvanishing real one form which annihilates the holomorphic tangent space to N. Let E be the line bundle of positive multiples of 0. Chern showed that associated to E is a principal bundle Y (a subbundle of the bundle of coframes of E) with an invariantly defined (up to CR equivalence) Cartan connection with values in
a u (n,1). Thus, in the real analytic
case, a complete list of local invariants can be given in terms of the local invariants of the Cartan connection.
In [34], Klembeck began studying the asymptotics of complete Kllhler metrics on bounded strongly pseudoconvex domains in C n. He showed that for a wide class of complete K~lhler metrics, the holomorphic sectional curvature of the metric tends to a negative constant near the boundary. Choose a defining function ~) for D which is smooth and strongly pseudoconvex on D. gi]- = - ~21°g(-~)/~zi0zT"
Define a Klihler metric g i f l z i ® dz~ by
(It follows from Fefferman's analysis of the Bergman kernel
that the Bergman metric satisfies these estimates for the metric and its curvature). Then g i ] - = I~i]/(--~) + q~id0j4~2 is a complete K/thler metric which blows up at the rate 1/(-~) v~ in complex tangential directions and 1/(--~) in complex ne.-'mal directions. Klembeck's result follows easily by explicitly computing the curvature tensor for the
48 metric and the tensor corresponding to constant negative holomorphic sectional curvature and comparing the highest order terms. This computation also makes it clear that estimates similar to the growth estimates for the metric hold for all higher order covariant derivatives of the curvature. This implies that the growth for any invariant tensor associated to the metric is well controlled on strongly pseudoconvex domains.
In [16], Cheng and Yau showed that on any smoothly bounded pseudoconvex domain in C n, there exists a unique complete Einstein- K~hler metric. Further, if the domain is strongly pseudoconvex, then the potential for the metric can be written as g =-log(-~) C°(D) ~
where ~) is a smooth defining for function for
D
which is in
c n + l ~ ) . It then follows from Klembeck's calculations that the holomorphic
sectional curvature for this invariantly defined metric approaches -1 at infinity.
Fefferman took up the study of boundary invariants in [22]. In this paper, he started with a smooth strongly pseudoconvex defining function ~ for D, and algebraically manipulated it to obtain a new function (which we will still call ~) ) which satisfied
the
equation
(-qb + Id012) I~i~l = 1 + 0(~) n+l.
00iT = ~2(~/~zi~zJ ' (~i j~k = ~k,
(In
this
notation,
[ d012 = (~id0 J (~i 7). Solutions to this equation automati-
cally satisfy the requirement that (-log (-0)) is the potential for a complete K~hler metric which is asymptotically Einstein to a certain order. Then, using coordinates (z,w) on D x C*, the function u = ~lw[ 2/n+l is the potential for a nondegenerate hermitian metric which restricts to a circle bundle over 3D as a nondegenerate Lorentz metric. The conformal class of this metric is biholomorphically invariant in the following sense: if the same procedure is applied to any biholomorphically equivalent strongly pseudoconvex domain, the resulting Lorentz metric can be pulled back by a biholomorphism of D 1 x C* =- D 2 x C* (which is induced by the biholomorphism D 1 ---D2) in such a way that the new Lorentz metric on the circle bundle is in the same conformal class as the original. Thus, this procedure results in an invariant conformal class associated to a circle bundle over the boundary.
Webster [48] used a simple construction to tie together Fefferman's approach to boundary invariants with Chern's approach. An alternative description of the relationship between the two approaches was described by Burns, Diederich and Schnider [11]. Their method had the additional advantage of providing an intrinsic derivation of Fefferman's conformal structure on the circle bundle.
We will follow Webster's
approach here because it is more closely linked with the extrinsic geometry of the
49 boundary, and the asymptotics of complete approximate Einstein-K~hler metrics.
Webster initially followed Fefferman's approach, beginning with a smooth strongly pseudoconvex defining function 0 and defining the potential for a smooth Klthler metric on D ® C* by u = 01 w 12/(n+l) as above. He next reduced the bundle of frames on 0D x C* to a principal bundle P of frames which are adapted to the boundary (adapted hemitian frames), and further to a subbundle
P, of unitary adapted
frames with a structure group K. Recall now C h e m ' s construction: over the line bundle E (of positive multiples of the annihilator of the complex tangent space), he constructed a principal bundle B~(=Y) of coframes.
Denote by B 1 the dual bundle of
frames with structure group G, and let B denote the larger bundle of adapted frames (not necessarily unitary relative to a hermitian form).
Webster exhibits a map f:
0D x C* ~ E which is the identity on the base OD, and associates to it a natural bundle map f:P --~ B which makes P into an S 1 bundle over B, and induces an isomorphism from G to K. He can then pull back the Cartan connection on B, which was described by Chern, and compare it with the hermitian connection on P1 associated to Fefferman's metric.
This procedure can be carried out when Fefferman's metric is
computed from an arbitrary strongly pseudoconvex defining function (~. When one assumes in addition that (~ satisfies (--~ + I d~ 12) I ~i~[ = 1 + 0(~3), then the curvature form for the Cartan connection pulls back to the curvature form for Fefferman's hermitian connection; moreover, the Cartan connection pulls back to the hermitian connection for some metric in the same conformal class as Fefferman's metric. All invariants from Chern's connection can be expressed in terms of invariants for the conformal class of metrics described by Fefferman. A byproduct of this result is that the intrinsic invariants of the boundary can be expressed in terms of the asymptotics of the complete Einstein Kllhler metric on D; however, the methodology is extrinsic in nature.
A converse to Klembeck's result on the asymptotic behaviour of metrics on smooth bounded strongly pseudoconvex domains was established in [6]. In this paper, it was shown that if M is an open complex manifold with a complete K~hler metric g, and if for some point p e M and some sector of the exponential coordinate system centred at p, the exponential map is injective, and the curvature and its covariant derivatives satisfy the growth estimates of Klembeck's metrics (expressed intrinsically in terms of distance from the point p), then there exists a natural boundary at infinity for this sector and the complex structure on the coordinate system extends smoothly to a strongly pseudoconvex CR structure on the boundary.
This boundary is natural in
50 the following sense: if q is any other point, and some sector of the exponential coordinate system centred at q is injective and intersects the sector from p, the overlap map is smooth out to the closure of the coordinate systems. Hence, the boundary of M and its CR structure can be defined independent of the base point p [8]. A byproduct of this result is that the CR structure on the boundary, and its Chern invariants can be expressed intrinsically in terms of the asymptotics of an approximate Einstein K~hler metric on the interior of the manifold. The next section will use the compactification of open complex manifolds to study the function theory on the interior. § 3: Function T h e o r y on Bounded Strongly Pseudoconvex Domains.
One natural approach to the function theory on open complex manifolds is via the 0-problem. In this approach, one starts with a smooth function with prescribed properties and modifies it by subtracting off the solution to an associated O-problem. The net effect is to take the orthogonal projection of the original function onto the space of holomorphic functions in some Hilbert space of square integrable functions relative to a weight function. The Hilbert space may be changed according to the desired properties of the resultant functions.
Siu and Yau applied this approach effectively to (n,0)
forms in [47] to produce functions of minimal growth; they then showed that such functions played the role of linear functions, and used them to define a biholomorphic map to C n (see § 1). However, no one has used this method successfully in producing bounded holomorphic functions.
On the other hand, Kohn [23, 35] has extensively studied the problem of solving the O equation on open complex manifolds M when M is a priori given as the interior of a compact complex manifold M with a smooth strongly pseudoconvex boundary. In his approach, he fixes a smooth hermitian metric on M, and considers solutions to an associated elliptic equation.
The formalism leads to a noncoercive boun-
dary value problem, which makes the estimates for smoothness at the boundary difficult to obtain. However, once the estimates at the boundary are obtained, then the solutions to the equation are smooth on M if the initial data is smooth on M, and in particular, the solutions are bounded.
In [23,35], Kohn solved the O Neumann problem in much greater generality than we presently need.
In order to simplify the exposition, we shall only present the
51 statement for complex manifolds with strongly pseudoconvex boundary. The manifold M will have a hermitian metric h. The square integrable forms on M form a Hilbert space, and ~-* will be the Hilbert space adjoint of ~r. We define the self adjoint elliptic operator [] = a ~* + ~r* ~. Let HP'q(M) be the square integrable (p,q) forms on M which are in the kernel of [] and let /"P'q(M) be the smooth (p,q) forms on M. Kohn establishes the following: (3.1) Theorem. Let M be a compact complex manifold with a smooth strongly pseudoconvex boundary and a hermitian metric h. Then 1)
For q > 0, HP'q(M) is finite dimensional
2)
HP'q(M) c hP,q(M)
3)
If oc e /,,o,1(~), i)o~ = 0 and oc is perpendicular to H°'I(M), then there exists a unique u e A°'°(M) such that a-u = c~ and satisfying the Sobolev space estimates Ilulls+l < C [[(xllsfor some constant C independent of oc.
H. Rossi and J. Taylor referred to a compact complex manifold with a smooth strongly pseudoconvex boundary as a smooth finite strongly pseudoconvex manifold. (It should be noted that in their definition, they required only that the integrable complex structure on M extend smoothly to the closure M. Thus, while the NewlanderNirenberg theorem guarantees the existence of local holomorphic coordinates at interior points, there is no a priori assumption made on the existence of such coordinates at boundary points.) Such a manifold is said to be complete if, in addition, it admits a global smooth strongly pseudoconvex function. In [46], Rossi and Taylor showed how Kohn's results could be used to prove an imbedding theorem for smooth complete finite strongly pseudoconvex manifolds. (3.2) Theorem. Let M be a smooth complete finite strongly pseudoconvex manifold. Then
there
exist
holomorphic
functions
fl . . . . .
fN ~ A0'0(~)
such
that
(fl ..... fy): M '+ cN is an imbedding. Their proof proceeded as follows. First, the global strongly pseudoconvex function and the technique of Hbrmander's weighted L 2 spaces can be used to show that dim(H°'l(M)) = 0. Thus, Kohn's theorem states that if o t e A°'I(M) and O-c~= 0, then there exists u e C~(M --) such that ~ru = c, and IlUlls+l < eliot[Is for some constant depending on
C
s,M but not on co. Rossi and Taylor then used the same technique
which Boutet de Monvel [10] used to imbed compact strongly pseudoconvex CR manifolds. Applying the technique in this situation and using Kohn's estimates, they were able to establish the existence of local coordinate systems for all boundary~ points;
52 more precisely, for any p ~ aM = M'xM, there exist holomorphic functions fl . . . . . on M such that (fl . . . . .
fn
fn) serve as local coordinates near p. Moreover, by amal-
gamating a sufficiently small neighbourhood of the image of p to the complex manifold M, they can consider p as an interior point of a larger finite complete strongly pseudoconvex manifold. Hormander's techniques for Stein manifolds can be applied to M to show that for any compact K c M, the algebra of holomorphic functions on K can be uniformly approximated by functions which are holomorphic on M. In particular, although a priori, it is only known (by the Newlander-Nirenberg theorem on the existence of local coordinates for integrable complex structures) that local holomorphic coordinates exist, the uniform estimates show that the local coordinates can be assumed to analytically extend to M (although not necessarily as global coordinates). Further, since the Steinness of M
implies that for any compact subset K c M,
bounded holomorphic functions separate points of K, if follows that global holomorphic functions of M separate interior points of M. By the above comment (that for any bundary point, there exists a larger finite complete strongly pseudoconvex manifold for which it is an interior point), the global holomorphic functions on M separate points on M. The result then follows by the compactness of M. The result of Kohn and Rossi-Taylor were used in [8] to provide an intrinsic function theoretic characterization of those open complex manifolds which are biholomorphically equivalent to a smooth bounded strongly pseudoconvex complex submanifold of C N. Before stating the result, we make the following definition.
(3.3) Definition. Let ~) be a smooth bounded strongly pseudoconvex exhaustion of an open complex manifold M. Then ~ is said to be smooth and strongly pseudoconvex at infinity if there exist constants C k such that." (i)
] log I d(~ I ] _< C o outside a compact subset
(ii) I1 vk~) 11< Ck (iii) I1 V k R* II -< Ck
where all norms are taken with respect to the finite K~lhler metric #i7=
02~/3Zi 3Zj=.
vk~) and V k R* refer to the kth covariant derivatives of ~) and R* respectively, and R* is the curvature tensor of the metric ~i]~ That is, ~) is said to be smooth and strongly pseudoconvex at infinity if the tive to this metric.
metric
(~i~ has bounded geometry and ~ is smooth rela-
53 We are now ready to state the theorem established in [8]:
(3.4) Theorem.
Let M be an open complex manifold. Then M is biholomorphi-
cally equivalent to a smooth bounded strongly pseudoconvex submanifold of C N ~.~ M admits a smooth bounded strongly pseudoconvex exhaustion t) which is smooth and strongly pseudoconvex at infinity.
The idea used in establishing the above theorem is to use the function ~ to attach a boundary.
Since d~ e 0 outside a compact,
M
has a natural collar structure.
Define T to be the vector field which is normal to the level sets of 0 (relative to the metric ~ij) and normalized such that T(~) -= 1. The collar structure can be defined by following the integral curves to the vector field T. This makes the complement of a compact into a product manifold, and defines a natural smooth structure on the boundary. The bounds on the curvature and the covariant derivatives of ¢ can be used to show that the metric ~)iT extends smoothly to a nondegenerate metric on the boundary, and the complex structure extends smoothly to a strongly pseudoconvex CR structure on the boundary. The resulting compactification is a complete finite strongly pseudoconvex manifold with a smooth K~thler metric ~)i~" The imbedding results of Rossi and Taylor apply to complete the proof.
§ 4: Weakly Pseudoconvex Domains The state of knowledge for weakly pseudoconvex domains is much less advanced than for smooth bounded strongly pseudoconvex domains, and it has been a well recognized fact that the weakly pseudoconvex case is in general much more difficult to understand. From a philosophical viewpoint, this can be understood in the fact that the weakly pseudoconvex case is the borderline between the good case ( smooth and strongly pseudoconvex) and the bad case (not pseudoconvex), and the closer that one is to the strongly pseudoconvex case (for example, domains of finite type), the easier the analysis becomes. A mathematical understanding of the problem in relation to the 0- problem is that this is the point at which uniform positivity of the Levi form is lost, and the subelliptic estimates at the boundary become weaker. On a still more rudimentary level, one no longer has a smooth defining function ~ for the domain for which ¢i]-is strictly positive.
Recall, for instance, that this was the starting point for
54 Klembeck in determining the asymptotic behaviour of the curvature for the complete K~thler metric (-log (--0))i3-; similarly, Fefferman's algebraic manipulation to obtain an asymptotic Einstein Klthler metric relied on the fact that ~i~ was uniformly invertible. An offshoot of this is that weakly pseudoconvex domains do not possess complete K~ihler metrics with the asymptotic growth properties on the curvature tensor satisfied by Klembeck's metric; even the metric, in general, satisfies different asymptotic growth estimates. On the other hand, much of the above program can be carried out, and while the results are still not complete, the outlook is promising.
The weakly pseudoconvex domains which are most completely understood are those of finite type. Many different definitions have been suggested for the concept of finite type; all such definitions attempt to express the degree of degeneracy of the Levi form on the boundary, or the maximum order of contact of analytic subvarieties with the boundary - that is, in some respect they refer to the degree of "flatness" of the boundary.
For a survey of the various definitions, their interrelationships and their
relationship to the O-Neumann problem, see D'Angelo's article in these proceedings [19]. We will mention here only the definition due to D'Angelo: a point p ~ 0D is said to be of finite type if the order of contact of all complex analytic subvarieties with OD at p is bounded.
Catlin [14] introduced a more general notion for weakly pseudoconvex domains than that of finite type. He said that a pseudoconvex domain D c c C n satisfied condition P if for every c > 0, there exists a smooth, pseudoconvex function ~ on D such that 0_<(~_< 1 on D and [~)ijq>_C[~ij] on ~D. In [14], Catlin showed that if D c c C n is weakly pseudoconvex of finite type (as in the sense of D'Angelo), then D satisfies condition P.
With these preliminaries out of the way, we will now summarize what is known for weakly pseudoconvex domains. As in the strongly pseudoconvex case, we begin with the biholomorphic invariance of the boundary. Fefferman's original result on the smooth extension to the boundaries of biholomorphic maps between strongly pseudoconvex domains has been greatly simplified and generalized using a variety of techniques (see e.g. [1, 2, 3, 4, 20, 41]). The most general result relies on an understanding of the Bergman projection operator P:
that is, the operator which projects the
space of L 2 functions on D orthogonally onto the space of L 2 holomorphic functions on D,
where the L 2 norms are with respect to the extrinsic Euclidean metric. A
55 m
domain D is said to satisfy condition R if P : C=(D) -+ C~(D). (By studying the formal setup to the i)-Neumann problem, it is easy to see that a domain D satisfies condition R if the ~- Neumann problem is globally regular on (0, 1) forms; that is, if the canonical solution u to ~-u = ct is smooth on D whenever ct is a smooth (0, 1) form on D). Bell showed that a proper map F • D 1 --+ D 2 between two pseudoconvex domains extends smoothly to a map F : D 1 --+ D 2 if D 1 satisfies condition
R [2].
Further, by an earlier result of Fornaess [24], if F is also a biholomorphism on the interior, then F extends to a diffeomorphism on the boundary. It will follow from the results of Catlin (described below) that if
D
satisfies condition P, then
D
satisfies condition R.
From a geometric viewpoint, complete Klihler metrics on weakly pseudoconvex domains will not admit the same nice asymptotic analysis permitted by Klembeck's metrics on the smooth bounded strongly pseudoconvex domain.
Indeed, since
Klembeck's metrics possessed such stringent asymptotic growth rates, it was possible to regularize the exponential map and to use Jacobi fields to deduce that the regularized exponential map yielded smooth boundary coordinates. In the general case, the asymptotic growth rates can be quite arbitrary, (see e.g. [7]) and the Jacobi equations will not provide the necessary estimates. On the positive side, Mok and Yau [43] established the following result: If D is a bounded pseudoconvex domain in C n, then D admits a unique complete Einstein K~ihler metric. (It follows from Yau's studies on harmonic function theory on complete Riemannian manifolds [49] that the Ricci curvature must be negative). Their proof involves exhausting the pseudoconvex domain by smooth strongly pseudoconvex domains, taking the limit of the Einstein K~hler metrics on the intermediate smooth domains, and obtaining an estimate to ensure that the limiting metric does not degenerate and remains complete. This proof outline is actually illustrative of a general principle for dealing with the remaining cases: first, solve the problem in the good case to obtain an canonically defined solution, and then obtain uniform estimates to ensure that the solution does not degenerate in the limit.
Recently, there has been much progress made in understanding the function theory on weakly pseudoconvex domains. Before describing the main results, we take note of two earlier results due to Kohn and Nirenberg. In [40], they showed that if the quadratic form QO,~) = I[~[]2 + 118 ,l[ 2 + I]-8"01[2 is compact on (p, q) forms, then the Er Neumann problem is globally regular on (p, q) forms. In the same paper, they also showed that if a Neumann problem satisfies a subelliptic estimate on (p, q) forms
56
(llcbll~ ~ c Q(~, ~) for some constant c independent of ~, where II 1[2 is a Sobolev cnorm, e > 0), then local regularity holds for the ~r Neumann problem for (p, q) forms; that is, the canonical solution u to 3ru = (x is smooth on D wherever c~ is smooth.
Catlin [12, 13, 15] studied the relationship between conditions on the boundary of a weakly pseudoconvex domain D and regularity for the O-Neumann problem. He showed that a subelliptic estimate holds for the ~--Neumann problem on (p, q) forms if and only if the domain D is of finite q type (D is of finite q type if the order of contact of the boundary with all q dimensional analytic subvarieties is bounded). It follows from the work of Kohn and Nirenberg that the O-Neumann problem on (p, q) forms is locally regular for domains of finite q type. In [14], Catlin also obtained some results for pseudoconvex domains satisfying condition P. He showed that if D satisfies condition P, then the form Q satisfies a compactness estimate on (0,1) forms; again, it follows from the work of Kohn and Nirenberg, that for smooth pseudoconvex domains satisfying condition P, the ~r-Neumann problem is globally regular for (0, 1) forms.
Kohn [37] studied the O-problem on general smooth bounded pseudoconvex domains using weighted Hilbert spaces. He was able to show that if D c c C n is a smooth pseudoconvex domain and if a is a ~3 closed (p, q) form which is smooth in D(q >_ 1), then for every k > 0, there exists a (p, q-l) form u which is in c k ~ ) and satisfies 3ru = c~. In addition, for every fixed integer k, the solutions u k satisfy uniform Sobolev estimates []u][s < C []o~[Is (s < k) with constants independent of c~. Finally, in [38], Kohn showed that by using a diagonalization process suggested by H~Srmander, one could find a solution u e C=(D) to 0-u = c~.
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40. Kohn, J.J. and Nirenberg, L., "Noncoercive boundary value problems," Comm. Pure Appl. Math. 18 (1965), 443-492. 41. Lempert, L., "La m~trique de Kobayashi et la rkpresentation des domaine sur la boule," Bull. Soc. Math. France 109 (1981), 427-474. 42
Mok, N., Siu, Y.T. and Yau, S.-T., "The Poincark-.Lelong equation on complete K~hler manifolds," Compositio Math. 44 (1981), no. 1-3, 183-218.
43. Mok, N. and Yau, S.-T., "Completeness of the K~hler Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions," Proc. Sym. Pure Math., Vol. 39, Part I, Providence, R.I. (1983), 41-59. 44. Narasimhan, R., "Imbedding of holomorphicaIly complete complex spaces," Amer. J. Math. 82 (1960), 917-934. 45. Remmert, R., "Sur les espaces analytiques holomorphiquement s~parables et holomorphiquement convexes," C.R. Acad. Sci. Paris 243 (1956), 118-121. 46. Rossi, H. and Taylor J., "On algebras of holomorphic functions on finite pseudoconvex manifolds," J. Functional Anal. 24 (1977), 11-31. 47. Siu, Y.T., and Yau, S.-T., "Complete K'~hler manifolds with nonpositive curvature of faster than quadratic decay," Ann. of Math. 105 (1977), 225-264. 48. Webster, S., "K'dhler metrics associated to a real hypersurface," Comment. Math. Helvetici 52 (1977), 235-250. 49. Yau, S.-T., "Harmonic Functions on complete Riemannian manifolds," Comm. Pure Appl. Math. 28 (1975), 201-228. University of Toronto Toronto, Ontario, Canada M5S 1A1
A CHARACTERIZATION
O F C'P n B Y I T S A U T O M O R P H I S M
GROUP
J. Bland
by T. Duchamp
M. Kalka
University of Toronto
University of Washington
Tulane University
§ O. I n t r o d u c t i o n . In this note we extend the results of Greene and Krantz [ G K ] to the case of compact manifolds. To be more precise, let M be an n-dimensional complex manifold, let p be a point of M and let G be a group of biholomorphisms of M fixing p. By differentiation the group G acts linearly on the complex tangent space Mp and, therefore, on the complex projective space CP(Mp) of complex directions at p. If the action of G on CP(Mp) is transitive then G is said to act transitively o~ complex directions. Our main result is the following theorem, which answers affirmatively a conjecture of Krantz: 0.1. T h e o r e m . L e t M be a compact complex manifold and suppose that there is a compact group of biholomorphisms of M fixing a point p and acting tra12sitively on complex directions at p. Then M is biholomorphic to CP n. 0.2 R e m a r k s . i) In the case where M is not compact Greene and Krantz show that M is either biholomorphic to the unit ball in C n or to C ''~ itself. Although not in the statement of their results, they show that the biholomorphism can be made to send p to the origin and to identify G with a linear subgroup of the unitary group U(n) with its usual linear action on C'n and that this biholomorphism is unique up to composition with a unitary transformation. tl) In [ G K ] the apparently stronger assumption, that G acts transitively on real directions at p, is made. However, this is unnecessary by the following argument: Since G is compact, we can arrange for G to act isometrically with respect to a Hermitian metric on M. Now let Sp be the unit tangent sphere to M at p. Then G acts on Sp and the Hopf m a p Sp ~GP(Mp) is G-equivariant. Assume that G acts transitively on complex directions; then its action on the unit sphere is transverse to the fibers on the Hopf map. Either G acts transitively on Sp (and the result follows) or the orbits of G cover CP(Mp). However, the second case is impossible since CP(Mp) is simply connected so such a cover would give a section of the Hopf map; but it is well known that none exist. ill) Since the space CP(Mp) is connected, the connected component of the identity in G will act transitively on complex directions whenever G does. Therefore, without loss of generality we may assume that G is connected. iv) After this work was completed we learned that the above theorem is a weaker version of a theorem of E. Oeljeklaus [O]:
Let M be a compact complex manifold and G' any group of biholomorphism on M. I f there is a point p E M fixed by G' and a neighborhood Up in M such that Up \ {p} is
an orbit of G' then M is biholomorphic to GP n. Oeljeklaus has also informed us that he and A. Huckleberry have obtained classification theorems for complex manifolds equipped with group actions by biholomorphisms
61 (see [HO] and the references therein). A c k n o w l e d g e m e n t . We wish to thank S. Krantz for posing the problem of characterizing G'P n by its group of biholomorphisms to us, J. Morrow who explained his work on compactifications of G'~ to us and E. Oeljeklaus for informing us of his work and the work of tIuckleberry.
§ 1. Outline of the Proof. In this section the proof of the main theorem is outlined. For the remainder of the paper M will be an n-dimensional compact complex manifold with G a compact group of biholomorphisms fixing a point p and acting transitively on complex directions at p. We will give M a Hermitian metric with respect to which G acts by isometries. Let Cp denote the cut locus of p (see [K] for elementary facts concerning cut loci). The results in IBM] show that if M is obtained from G'n by attaching a copy of G'P '~-1 then the resulting manifold is biholomorphic to G'P •. Hence, it suffices to show that M\Cp is biholomorphic to G''~ and that Cp is a complex hypersurface which is biholomorphic to C p ~-1. The first step in the proof, presented in section 2, is to show that Cp is a compact, connected, complex hypersurface whose complement is biholomorphic to C'~. The next step proceeds as follows. The biholomorphism between MkCp and C '~ allows us to define a singular holomorphic foliation of M \ Cp (just take the complex lines in C n passing through the origin). The leaves of this foliation are holomorphically parameterized by their (one dimensional) tangent spaces at the point p, i.e. by points in GP(Mp). We show that the closure of each leaf is a smooth complex curve, biholomorphic to G'P 1 which intersects the cut locus transversely at a single point. The closures of disjoint lines will be shown to be disjoint. This foliation furnishes the biholomorphism between CP(Mp) and Cp needed to apply the results of [BM] and completing the proof of the theorem.
§ 2. T h e c u t locus o f p. We begin the analysis of Cp with some elementary observations. By Remark 0.2.ii, G acts transitively on the unit tangent sphere at p. It follows that G acts transitively on the set of geodesics starting at p. Since each geodesic from p intersects Cp, it follows that G acts transitively on Cp and that the distance from p to any point in Cp is a constant which we will choose to be 1. In particular, the cut locus, being a G-orbit, is a smooth, connected submanifold of M. Let expp be the exponential map at p, Bp the unit ball in Mp and Sp the unit sphere in Mp. Then expp restricts to a map from Sp to Cp which, by trailsitivity again, has constant rank. The following lemma summarizes several important properties of the exponential map and is essentially contained in [K], p 100. 2.1 L e m m a . The fibers of the map expp : Sp ~ Cp are in 1-1 correspondence with the geodesics joining p to a fixed point in the image of expp. The manifold M is obtained by attaching the ball B; to U; via the above fibration.
62 2.2 P r o p o s i t i o n . The cut loctzs of p, Cp, is a complex hypersurface of M whose complement is biholomorphic to C ~. P r o o f . By [GK] the complement of the cut locus is biholomorphic to the unit ball or to complex n-space and, therefore, supports non-constant holomorphic functions. Since Cp is a real submanifold it has pure Hausdorff dimension. Therefore, by the result of [S],Lemma 3(i), if Cp had real codimension greater than 2 then M would support nonconstant holomorphic functions. However, M is compact; hence Cp has real codimension either 1 or 2. Suppose that the real codimension of the cut locus is 1. Then the map Sp ~ Cp induced by the exponential map is a finite covering map. But the number of sheets is precisely the number of geodesics starting at p and terminating at a point q in Cp. The geodesics joining p to the cut locus are extremals of the arc length functional with variable end point on Cp. As such they satisfy the transversality conditions of the calculus of variations, i.e. they intersect Cp at right angles. Since Cp is a real hypersurface the covering must be 2 to 1. Now a fixed point free action of g2 on an odd dimensional sphere is orientation preserving (since the Z2 action has no fixed points it follows from the Lefschetz fixed point theorem that the action of Z2 on the top dimensional homology group is trivial), hence, the cut locus is orientable. On the other hand the normal bundle of the cut locus is one sided by the following argument. By transitivity of the action of the connected group G on the sphere, Sp, any two geodesics starting at p can be connected by a family of geodesics starting at p, showing that Cp is one-sided. Because there are no one-sided, orientable hypersurfaces in an orientable manifold the codimension of Cp cannc.t be 1 and is necessarily 2. By [GK] the complement of the cut locus is biholomorphic to complex Euclidean space or to the ball. Since the ball supports bounded, non-constant holomorphic functions and by [S],Lemma 3(ii) such functions would extend to all of M contradicting the compactness of M, the complement of Cp is biholomorphic to C '~. It remains to show that Cp is a complex hypersurface. To see this let G c denote the complexification of G. Then G c acts holomorphically on M and the orbits of G c are complex submanifolds of M. Since the complement of the cut locus can be identified with complex n-space in such a way that G is a subgroup of U(a) it follows that G c leaves the complement of the cut locus invariant. This in turn shows that G c leaves C~ invariant. The group G acts transitively on Cp therefore Cp is on orbit of G c and a complex submanifold of M. Q E D
§ 3. A foliation of M by UPl's. In this section we present the proof of Theorem 0.1. The idea is to construct a foliation, 5, of M by C P I ' s which intersect Cp transversely and to use the foliation to construct a biholomorphism between Cp and C P ~-1. Theorem 0.1 then follows from the results of
[BMJ To construct the foliation }" start with the observation that since M \ Cp is biholomorphic to G''~, it is naturally foliated by lines (identify p with the origin and consider the
63 foliation of C'n by lines through the origin). This foliation is singular at p, but becomes smooth after blowing up p. We will show that the closure in M of each line is smooth and intersects Cp transversely in a single point. The foliation 5r is then the foliation whose leaves are the closures of the above lines. That ~ is holomorphic is clear because it is holomorphic on the open dense set, M \ Cp. To construct a biholomorphism between the complex projective space CP(Mp) and Cp just map the complex line [v] c Mp, v E Mp to the point of intersection of Cp and the leaf of 5r whose tangent space at p is Iv] 3.1 R e m a r k . The biholomorphism between C '~ and M \ C~ can be extended to a biholomorphism between C,P n and M without using the results of [BM] as follows. Let q be any point in the hyperplane at infinity, let [vq] c G'n be the unique line whose closure in C'P '~ contains q and let Lq be the leaf of 5r whose tangent space at p is Iv] (recall we are identifying Mp and Cn). The image of Q is defined to be the unique point in the intersection Lq N Up. The proof that the closures of the above mentioned complex lines intersect the cut locus transversely and in a single point requires the introduction of an intermediate foliation. To construct a leaf consider any non-zero vector v in the tangent space Mp. (For convenience we will identify M \ Cp with C'~ and Mp with C '~. This abuse of notation will cause no confusion and simplifies the notation.) Let K v be the subgroup of G which leaves v fixed and Vv the linear subspace consisting of all vectors fixed by K v . Note that if w is in Vv than the geodesic t -+ expp(tw) is left fixed by K v . (Because G acts linearly on Vv we may identify the tangent space of Vv at the origin with Vv itself.) Therefore, the vector space Vv is a union of the geodesic segments t ~ expp (tw), 0 < t < 1 , w E Vv and the closure of Vv , which we will denote by the symbol Nv, is the union of the geodesics t ~ expp(tw), 0 < t , w E Vv • We claim that the complex dimension of Vv is either 1 or 2. Clearly, because Vv contains the line generated by v, its dimension is at least 1. To see that the dimension of Vv cannot exceed 2 proceed as follows. It is easy to see that for any h C G, the conditions hv = v and g h g - l g v = gv are equivalent. The next two equalities follow immediately:
Kgv = gKvg - t and
(3.2)
Vgv = g(vv)
If H v is the subgroup of G which leaves the space Vv invariant then above equalities can be used to prove the chain of equivalences
g ~ Hv ~
g(Vv)=Vv
~
Vyv = V v
~
Kgv=Kv
~
gKvg -1 = K v .
The last equivalence shows that K v is a normal subgroup of Hv. Further, for any automorphism g E G such that g - i v C Vv the condition g-1 E / I v is easily seen to hold. This gives the following characterization of Hv:
64
(3.3)
Hv - {g e C l g - ' v e Vv}
From (3.3) and the assumption that G acts transitively on directions it follows that H v acts transitively on directions in Vv and that the group Hv/Kv acts transitively and freely on the unit sphere in Vv. But then the group Hv/Kv is diffeomorphic to a sphere and therefore either the one or the three sphere (since by linearity of the G action on O n, Vv contains the complex line generated by v, Hv/Kv is not the zero sphere) and the complex dimension of Vv is either 1 or 2. Observe that since K v is normal in H v formula (3.2) shows that given two vectors v and w the spaces Vv and Vw are either disjoint or equal. We next show that the set N v is a smooth, compact, complex submanifold of M which intersects Cp transversely and that the family, Nv, v 6 Mp forms a smooth foliation of M which is singular only at p. Since Vv is a complex analytic submanifold, its closure will automatically be complex analytic if only we show that is smooth. First note that since the Lie group Hv, and hence the connected component of the identity in Hv, acts transitively on the unit tangent vectors to Nv at p, it acts transitively on the intersection Nv n Cp. Therefore, N~ intersects Cp in an orbit of a connected Lie group, hence, the intersection,Nv R Cp, is a connected submanifold. To show that N v is a submanifold it is sufficient to show that it is the union of all geodesics which intersect Nv R Cp and are normal to Cp. To see this let q be a point on Nv n Cp, let t --~ expp(tw), w ~ Vv be a geodesic containing q and let Wq be the tangent vector to the geodesic at q. Since every element of K v fixes the geodesic (and therefore Wq) it follows that K v fixes all normal vectors to Cp at q fixed (this space is one dimensional and has wq as basis). But K v acts by isometries and therefore fixes all geodesics normal to Cp at q. From the definition of Vv it follows that these geodesics are all in Nv, as was to be shown. To see that the family of all forms a foliation it suffices to show that for v, w E Mp, the submanifolds and N w are either equal or intersect in the single point p. If and N w intersect at a point in M \ Cp then they intersect along the unique geodesic joining the point to p and following this geodesic to a point in Cp and therefore intersect Up. But the analysis of the above paragraph shows that if and N w intersect in Cp they are equal. The manifold M is obtained from the complex one or two dimensional vector space, Vv, by attaching the smooth, compact, connected complex manifold N v N C~. If the dimension of Vv i8s one then we have already shown that the closure of each line in M \ Cp containing p is biholomorphic to O P 1 and that the closures of each line intersects Op transversely in a single point. If the dimension of Vv is 2 apply the result of Morrow ( [ i ] , L e m m a 5) to conclude that Nv is biholomorphic to O P 2 and identify Nv N Cp with the O P at mfimty m C P . The closures of complex lines through the point p in O P 2 are all smooth OP~'s which intersect the hyperplane at infinity transversely in single points and form a foliation of O P 2 which is singular only at p. Because the intermediate foliation intersects the cut locus transversely so does the foliation of M by O P l ' s . 1
•
•
•
2
65
§ References. [BM] L. Brenton and J. Morrow, Compactifications of C n, Trans. AMS, 246(1978), 139153. [GK] R. E. Greene and S. G. Krantz, Characterization of complex manifolds by the isotropy subgroups of their automorphism groups, preprint, (1984). [HO] A.T. Huckleberry and E. Oe[jeklaus, Classification theorems for almost homogeneous spaces, Institut Elie Caftan 9 n °, January, 1984. [K] S. Kobayashi, On conjugate and cut loci, Studies in global geometry and analysis, S. S. Chern ed., MAA Studies in Mathematics Vol. 4, 1967.96-122 [M] J. Morrow, Minimal normal compactifications ofC ~, Rice University Studies59(1973), 97-112. [O] E. Oeljeklaus, Ein Hebbarkeitssatz fiir Automorphismengruppen kompakter Mannigfaltigkeiten, Math. Ann. 190(1970), 154-166. [S] B. Shiffman, On the removal of singularities of analytic sets, Michigan Math. Journal
15(1968), n1-120.
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~,
f
. . . . . 0)
: ~n
÷
~k
< p
exists < n
Then
the
=
1
of
there
is
coincides
key
step u
and
~
a
such
set that
K
of
for
vectors
that
lemma
set
It
=
there 2
{ D 2 f ( u ) (v ~, ,v p)
u { D f ( u ) ( v J ) ] jn= 2 u
Ck
1201
(b).
vj
a mapping
with
f
F,
on
{
is
S
an
n N
(I~0-~] <
with
p)
~ K} on
n N. n
application with
: (Z,p)
holomorphic
~
--
Proof.
of
Theorem.
3~ k
Further,
independent.
values
open
(~,p),
{f(u) ]
N
of
involving
< n(n+l)
integers
the
finishes
and f
where
and
(u I . . . . ,u n) .{{ S
N
n
1
setting
=
Assume C2(~
>
generalization
~0 u
n (]~J
of
Cramer's
rule.
{ N.
Consider
for
>_ l)
and
k
2
< n
the
equations <w,f(!u)> <w,Df(~
:
1
u) ( v J ) >
=
0
<w,D2f(_l--u) ( v ~ , v P ) >
By
Cramer's
rule
holomorphic Thus
f
that
the
proper
on
of
a
function
N
•
cn). 6 we
theorem
If
will
But
f
to f
a
on of
(~,p) w
through larger
=
F(~u)
the open
which
slice set.
is
implies
that
several
complex
variables
to
a
continuation
proving
this
~
K
provides
determined It
slices
prove
prior
0
function
of
as
mappings.
4.5.
a
extension
section
following
Theorem
an
holomorphy
holomorphic
In
is
continuation
has
function
there
=
< j
this
theorem
a
by
routine
SoU. to
extension (i~e.
as
show is
a
for
rational the
result
we
proved
then
f
is
[2].
f
~
I
2
(n,n+l)
and
n h
3
equivalent
70
to
a mapping
p
q
is
~, w h e r e q a polynomial, of
degree
Note not
that yet
section
most valid
~k
are
"
n and
We
of
pose Bk
k,
there
is
and
proper
I I zl I 2N
by
the
obtain By
obtain
a proper
proper
example
maps
D'Angelo
p < 3
4.1).
and
It
this
question
for
the
~
for
proper
of
Assume f
is
the
maps
of
arbitrarily
write
B k.
degree
i ~ and
k
is
f?
high
does in
each
depends
ideal balls.
For
Note
by
term
map fixed
that
degree
on
n+l from
a proper
Then,
calling
the
dimensional
of
I Izl I 2N
Here
modulo
f
÷ n maps
a polynomial.
on
to
lower
stated
question.
theorem,
f
has
applicatfon
0f
obtains
proper
the
We
one
by expanding I fj I 2,
both
n
and
I Izl I 2 - i,
one
can
This
one
obtains
is
how
all
conjectured
~utomcrphisms
that
by
~f b a l l s
holomorphic
combining in
mappings
this
intermediate that
extend
process
with
dimensions, holomorDhically
boundary.
end
this
section
with
an
interesting
tool
that
has
been
useful
work.
Theorem
4.6.
n-i
Assume
f
dimensional
dimensional
subspace
satisfies affine
D
of
f Proof.
Fix
to
bound
expansion to
are
component
upper
map
the
the
we
degree
(Theorem
return
restrictions
following
Namely,
one
an
of
(3.5).
John
is
Webster We
results
multinomial
reducing
our
of
Faran.
polynomial
increase.
in
theorem
suitable
each
an
k
past
mapping
< p-l.
of
these
with the
into
produce
N.
the
result
letting
we
a polynomial
6.
B
can
the
they
n +
n
improves
Although case
of
this
improve
is
Since
may
assume
u E N
a unit
and
vector
notation
(vj)
loss
consider
orthogonal £. ] =
Ck
" A
automorphisms without
the
conditions
subspace
of
such
map
affine
subspaces that
pairs
(vj,~j) ~ j
to
and
~. 3
£. 3
times
(v.,...,vj), 9
Lemma
4~4
and
exists
a k-i
affine
subspaces
n IB k
generality
u
of There
that
n Bn ~ D
of
C n.
into 0 ~ A
and
1,2~...,k
is is
function
D ~ l f ( t u ) (v 1) g( k ) = det D~kf(tu)(vk )
a positive used.
The
f(0)
where integer~
= 0 . vj
is The
holomorphic
71
is
zero
on
arc
of
the
= span{DJf(0)
D
has
an
dimension
unit
(v) ~
< k-l.
But
f (w)
circle.
; ~ = the
Hence
g(%)
1 , 2 , 3 .... , < u , v >
power
series
~ 0
and
= 0}
expansion
=
j=l shows
that
the
dimension
Results
5.
on
Several [6,7].
They
higher
range
of
f
on
the
space
{v
~ B
:
n
=
0}
is
of
k-l.
discontinuous
papers
have
study
proper
dimensional
case
mappinKs.
been
done
by
maps
from
Low
[Ii]
~n
÷
E.
Josep B 1
has
Globevnik
into
Bn
and
L.
Stout
, n _> 2.
established
the
In
the
following
result. Theorem
5.1.
> 0,
R
h
Let
< i.
which
is
~
This
a significant
z ~ S
is
there
are
two
function
(and
appearing
in
N
this
For there
Hakim
=
N
the
of
The
the the
FI,...,F N
n
let
; j =
+ h(z) I = 1
computational wish
us
h.
with
on.
technique the
in-
z,< ~ S ( = ~ n
of
radius
result
such
but
comment
on
For z0
covering
balls
to
comment
about
= N(n,~
of
If(z)
approximation
mapping ball
N
we
the
following
that
highly
that
First~
and
< R).
is
proof
mapping
mapping
such
n I~I
applying
integer
{B(zi,j;er)
B ( z i , j ;~r)
the
denote
an
a continuous
•
proof
[8],
B(z0,r). exists
in -when
by
range
and
families
Fi =
in
be
(nonconstant)
The
Sibony
in
by
there
exist
and
Iz-~I//2
metric
~ > 1
result.
constructed
teger
6(z,~)
lh(~) I < ~
involved
by
a
holomorphic
ideas is
B2N
exists
h
developed
set
: S = there
: Bn for a l l
The
C 2N
f
Then
is
that
radii
for
)
r
known. r > 0
~r
1,2 .... ,N.}
B(Zl~k;~r)
= ~
j # k
and S =
We
do
very
not
know
useful The
Let
{E.; J
let
Tw
to
second
U i,j of
B(zij'r)
any
good
obtain tool
such that
j = 1,2,...,2N} be
the
complex
estimates
on
the
size
of
N
and
it
would
is
the
following.
be
estimates. is
important
be tangent
the
in
usual
space
at
this
proof
(canonical) w.
Then
basis Tw
in
n B2N
C 2N
is
and
a
72
ball
of
1 <
1 - ' I lwl ' ' I' 2
radius
i < N
(in
2N-I
space).
For
w
~
~2N
and
define
n.l (w)
=
n (w)
=
if
E2i-i
]w2i]
Jw2i_ll
+
: 0
and l
w2iE2i_l
- w2i_l,E2i
rr~2iE2i_l otherwise.
These
vector
fields
IFni<w)[i
observation
If
I
is
]B2N
a
set
with
of
=
For
<_ 6 ni
b
I c and
c Tw,
step
i#k
0
0
indices,
vectors induction
=
this.
lw - wi]
orthogonal initial
is
satisfy
= 1
Low's
- ~2~_iE2ilr
>
0
there
is
{I,2,...,N} [wl ,
i ~
involves
I, a
and
lwi[
>_ b
such
that
close
6 >
0
w,w i
then
(i
there
In i
scrutiny
such e
I)
are
in
exist
- hi(W)
of
that:
the
I <
s.
The
function
N
g(s)
gi(z)
~ i=l
where
N gi(z)
The
i ~ ~- . P i j ( z ) n i ( f ( z i j j = l iJ
:
scalars
1/2
i - If(~ij)F
the
functions
are
Pij(z)
P i j (z)
Low
has
results
domains. of
this
sults the
We type.
are
ball
Theorem
similar
mention It
valid
is
for
=
to
scalar
exp(-m(l
Theorem
one
more
due
to
-
5.1
very
F.
bounded
valued
peaking
))
for
C
interesting
Forstneric
strictly
functions
strictly and
[5]
convex
2
and
pseudoconvex
pertinent although
domains
we
result his
refer
re-
only
to
situation. 5.2.
There
property.
If
such
[ Ihl ] 2
that
h
<
I,
is
: ~n
a
a
then
such
F = (f,h)
positive
÷ cP
extends
I [h(z) ] I ~
that
2
N
~J and
))
is as
there
: ~n
a is
inteser
N
a nonconstant continuous
maps
the
~n
following
holomorphic function
a holomorphlc
÷ C 2N÷p
with
mapping
properly
mapping
to
g f
into
and n ..... : ~n ÷ C2N
~2N+p"
73
A
significant
Corollar
X
5.3.
embedding
F
tinuously The
corollary For
: ~n
to
Low.
For on
holds.
to
integer
n ~
v.3
one
extend is
the
v. can J u.+iv. = e J J
Z =
of
Theorem
not
to
4.2.
J.
we
tinuous
can
to
we 4.3
1,2
harmonic
e
=
it
be
define
f ~
is
choose
~ n smoothness
5.3
the
G2
on
In
only
Professor n + ~2N
not
n +
which
in
S.
The
for
examples.
the
For
con-
those
used uI
that
the
by
and following
then
=
i
,
B I.
Then
gj ( Zl)
=
with for
mapping and
satisfies
the
hypothesis
corollary. in
this
in
same
direction.
a
this
is
an
~
it
in
I
can
shown
not
following
be
us
It
is
=
a
approaching
is
z~(l
~
4.4 not
occur
probably
2
z 2)
this
satisfies
to f
for the
the
=
n
S.
In
Corollary
functions of
a proper
continued unfair
it
G1
from
any is
of
open
fl(Z)
=
f(Z)/l
cap
interest
-i/2 and
and
[8]
map
over but
It
Lemma
holomorphic 2 the C
technique of
5.3. use
is
satisfies DB
extends
?
n Corollary
then
of
question•
which
n+l on •
difficult
f
subset
Using
o
observations.
* B
and
(n,n+l) if
con-
and
~ ~ B which n n+l expects t h a t t h i s is
for
construction
is
and
is a
~
from
n holomorphic
2
in
One
holomorphically
question
f(Z)
n
valid
dense
discontinuities
has
there
?
Lemma
4.4
is
open
problems
from
f
between
is
~ M(n)
mapping
to
mapping
result
on
some
technique
f
M
is h o l o m o r p h i c . . .
with
Lemma
if
every
{ (e i 8 , 0 , . . . ~0) } .
If
the
which . .
mapping,
1
obvious
Sibony
to
÷ ~n+2
a proper
see
that
curve
such
uj
Iz[
result
~M
contact
criteria
the
1 < ~
~n
holemorphic
fact
on
s
continuously
do
conclude
on
<
i.
construct
some
k =
of
to
functions
and
G j (z 1 . . . .. Zn)
section
a Lipschitz
n There
÷
n
this
a proper
as
to
we
n ~
: ~
extended
but is
holomerphic
extended
[6]. Let
to
similar
~I
continuously
another
g
possible
not
1
establishes
close
to
conjugate
This
5.4.
us
possible If
two
mention
mapping
Let Is
produces
extended
Globevnik
Proposition
be
2u 1 +
be
j =
Finally due
a ~roper
not
are
) ~ ~ ~ For 0 n n (Z,GI(Z) ,G2(Z)) maps
=
is
can
5.2
(Zl,...,z
h(Z)
there
F
continuously
2u I
gj
1
that
Theorem
harmonic
e and
such
prove
corollary
which
If
following.
~ ~n+2+2N
used
the
~i
each
the
n
techniques
u2
is
Ill I~
74
do
there
exist
a proper 6.
map
The
functions into
Earlier P --. q
we
6.1.
Proof.
We
know
pj
!
degree show
that
generality
showed
If
3.
If
is
we
can
q ~
B =
0.
0
on
The
and so
q ~
there
we
may
a linear
where I IVl ] =
i,
We
The Df(0)
(which
P
maps
V of
=
=
( V l , V 2) 13 )
=
(i,0)
-
= L
is q
bol
> O.
we
have
0
3)
i,
(z 2, ) -~i
inner
q <
q ( Z 0)
2 =
and 0
Without
q(e)
=
0.
we
can
loss
of
Thus
theorem is
for
the
similar
special
and
will
case appear
in
-
finite
set
has
maximal
nonsingular.
+
f(Z)>
T
Also we
% e C,
IV),
Q +
rank
may
off
by
a
composing
f
in
6.1.2
=
C2
III
~ 1
where
=
L,Q,T
implies
i
and
I IZ] I =
. are
3 respectively.
6.1.1
If
(by
homogeneous Z =
equating
( Z l , Z 2)
polyand
coefficients
0 one
,P(z2,
and
on
zero
concludes
TI(Z) (al0z2 must
exist
homogeneous
TI(Z)
(T
Zl)>
So =
Examining (TI,T2,T3),
- a01zl)
quadratics =
with
assume
a01z2,b01z2,0)
With
2 and
2 7z 2
+
unchanged)
+
then
z I)
product
in
6.1.3 There
a rational
that
the
is
2 yz 2
+
0)
f
(alOZl
= L +
degree
holomorphic
terms
3)
o__nn ~ 2
maps.
and
2)
the
mapping
leaves
6.1.2 is
(fl,f2,f
was
deg
that
Faran's
B #
Zl) (i
0 = Hence,
such
e =
and
of
f
holomorphic
where
e ~2
(with
then
is
Z0 =
2)
(PI,P2,P
f
of
L(Zl,Z
=
I2(2,3)
one
nomial
that
Thus
6.1.1 Write
Z0
establish
~2"
such
holds.
to
result
(i
> 0
f e
(l-Zl) (I-~zI-BZ
q(Z)
assume map
alO
=
~2"
on
$2
(PI'P2'P3) q
is
assume
paper.
0
on
then
f = -P q
equivalent
general
a forthcoming
if
following
I2(2,3)
that
q(Z) and
that the
f ~
f
f3
continuation.
However,
Theorem
and
~3 ?
holomorphic
mapping
f2
ZlRI(Z)
the
fourth
etc.)
- b01ZlT2(Z)
=
R1
with
and
R2
0
degree
75
(aloZ2
T2(Z)
Further,
6.1.3
- a01Zl)R2(Z)
implies RI(Z)
- b01R2(Z)
= 0
so that
For
a dense
R2(Z )
1 -bo1
set
Z
of
RI(Z) in
1
6.1.4
S
and
1 ( C,
III
= 1
~ 1
7 This
implies
Equating
coefficients
of
: s(xz)s(!z).
12 y i e l d s
-
-
TI(Z) (al0Zl Using
the r e l a t i o n s h i p
Rl(Z)(alolz
2
+ a01z 2) + T2(Z) (b01z2)
i12
for
+ [01ZlZ2)
R1
and
= (~z21 + yz 2 R2
in this
+ RI(Z) ( a l o i z 2 1 2
equation
_ a01zlz2)
= ~Zl2 + 7z
Thus 1 2 2 RI(Z ) = ~ l o ( C Z l + Yz 2) We have
now z
1 2 2 TI(Z ) = a ~ 0 ( a z I + yz 2) __
T2(Z) and
using
6.1.5
the
=
(z2 ID0
continuity
2
al0 al0b01 at
e
z2) (~ 2 + y Zl z 2)
(for ]B 2 ) one
sees
2 Zl , 2 2 al0z I + a01z 2 + a20z I + a l l Z l Z 2 + a02 z~ + - - < ~ z l + yz 2) al0 ± f (Z) = 2 (i - ~Zl) (i - z I) + 7z 2 2 zl(alO a l O z 1 )(l-z 1 ) + a 0 l Z 2 ( 1 - Z l ) + z 2 ( a 0 2 + a l O z 1) q z2 b01z 2 + b20z21 + b l l Z l Z 2 + b 0 2 z ~ + (b--~l
f2(Z)
:
q
a01 2 Z2z ) a l 0 b 0 1 Z l ) (~Zl + y
76
_
z2(bol
f
(z) 3
~
2
--z b01
1
) (l-Zl) +
I@I
curve
< 2
(a real
F(~,$,@)
number),
= F in
S
=
zI
Letting
r +
1
and
i
using
6.1.6
2 I) + Z l ( C 0 2 + Cl2Z I) +
@ real
-c
cos
f2(l,O)
the
c20 = i-~
f3 (I'0)
substituting letting
the c + 0
lim r+l
lim r+l
equations we
+ (a02
(l-~)e i$ + 7(2 " a01 e l $ alob01
+
0 < E < 2 cos
$
define
we
obtain,
i$
of
f
on
~2
6.1.5
and
fl(r,0)
f2(r,O)
f3(r,O)
for
F(~)
into
using
Y )(2 alO cos
cos
$)e 2iS
al0 alO
$)e 2i@ cos
i $)e 2iS
a01 =
+
(c02
+
cos
Hence
cos ~(alO
=
_
_]i_
al0
+
i -
alObol(l-~)
$)e 2iS
c12) (2 cos
(i - e)e i$ + 7(2
a02
e
= lim r+l
Y a 0 1 )(2 alobol
(b02
( l - ~ ) e i$ + 7(2 c 20 ei$
3 c03z 2
deduce
i$ al O )e
(alO-
2
continuity
~01
a
=
and
$ - ~
a l o b o l (l-a)
6.1.7
3
2
e
alO alO I - a
fl (I'0)
and
7
b --z ) + - - z Ol 1 b01
by
z 2 = /2s
By
al0
2 2+ c 3+ z2 2 3 = c20zl + CllZlZ2 + c02z2 30Zl c21 iz2 + C l 2 Z l Z 2 + c 0 3 z 2 q 2 c 2 0 Z l ( l - z I) + C l l Z l Z 2 ( l - z
For
7aOl
~a01 2(1- i) + z 2 ( b 0 2 al0b0]~z 1 z
$)e 2iS
$)e 2i$ ~' ) alO
c20 l-c~
6.1.6
a
77 YaOl
b02 c02 Using
this
= -c12
information
zl(alO - ~z al0 fl (z)
BaOl Y
+
alobo1
we
1
alobol(1-a) c207 l-a
+ can
) (l-z
1
rewrite
equation
6.1.5
as
follows
al0 . . . . y 2 al0 )a01z2 (l-z I) +'a~oZ2 (I -z 1 ) l-
) + yz~(
q(Z)
=
~a0l -----( aolb01
2(l-z ) +----fl~z2 + z 2 ( l - Z l ) Zl 1 (l-a) 2 )
f 2 (Z) =
(b01
~ 2 -+ z2(l-Zl)" - b 0 1 z)
q(Z) i y a01 ] r
c20(z~(l-Zl) f3 (Z) =
Since
f
is
+ _7__ l-~Z22 ) + C l l Z l Z 2 ( l-Zl ) _ C l 2 Z 22( l _ Z l ) q(Z)
i -C __(]B 2)
we have
f(e) - f(Z) ] - ~ ] .... as
+ CO 3Zl3
> D f ( e ) ( O , e l~)
s + 0, w h e r e Z
= (Zl,Z 2) = (I - s e
i¢
, /2 s cos
%
-
E
2
e
i? )
•
Since
(l-Zl) fl(l,O)-fl(Zl,Z2)
2( alO al0 i-~
=
alO
z ) l-z~ ~ Y z 2 ( l - z I) 1 aolZ2( l al0 z
q(Z)
and
l](1,o) we
- (zl,z 2 )112 = 2 ~ cos
conclude -aolei% e-i~Dlf(e)(O,e
i})
= Df
(e)(O,l) 1
: (l-~)e i% + y e 2 i 9 ( 2
cos
%)
78
-a01 (i-~) Therefor fl
either
and
f2
duce
to
a
have
the
y
must
=
0
or
have
quadratic
+
=
functions "
a01
=
a
0.
as
a
If
This
has
a
limit
and
we
are
factor.
CllZlZ 2 -
c12z2
on
the
as
~ ÷
0
reduced We
and
to
numerators
fl
and
function
of
f2
f3
re-
would
of
this
(i-
equation
eZl)(l
are
C1
- z I) functions
E2) 3 / 2 e 3 i ~
limit
% - e2
is
constant.
p
is
Therefore
quadratic
and
l -
f
alO
=
0
( l - z I)
01 2 b z Ol 1
az 1
clearly
is
2 Cl2Z 2
aOl
=
=
Cll
analytic
on
O,
=
alO
c12
~2
=
=
I~
b 01
=
1 and
I~I
>
1
0
(and
if
there
prove). Returning
to
the
case
y
#
z2(l-Zl)(b01
0
0 =
lim r÷l
( l - z I)
f2(r,0).
and
-- z ) + b01 l
= (i - ~ Z l )
=
=
a01z 2
CllZlZ 2 -
c20
f2(Z)
co%
S(Z)
1 - ~z 1
then
proves
t cos
where
bolzl ) +
=
I,
this
f = ~
al0Zl ) +
=
f3(Z
f2(e)
on
i - ~Z~
2 c20z I +
=
3 c03z2
have
=
f2(Z
So
Thus The
+
s cos % -
z2(bol
to
the
2
right
zl(al0 fl(Z
This
0
means
linear.
la[
=
factor.
linear
[(i - ~ ( i - ~ e i % ) ( ~ e i % ) ] / 2
If
y
1 - ~z I
ei%c03(2
is
i(2~-~)
~)e
value
f3(Z)
~2
cos
(l-Zl)
over
2 c20z I +
Both
y(2
+
2 Yz 2
Then
aOl ~ b01
=
z
3 2
0
we
have
is
nothing
79
i($+~) e D f 2 ( e ) (0,i)
(b01
b
n~I ) + ~ ( 2 b01 cos
$)e 2iT
as a c o n s e q u e n c e
(bol
£ ) + --7--(2 cos bol bol
(i-~) is a c o n s t a n t .
Since
(bol
- bo I
1 c = lbo
Hence,
+ 7(2
y # 0
cos we
1 bol
b01
- bo I
= 1
and we
_
slice
(0,z2)
of
the m a x i m a l fl(0,z2)
+
~ b01
-
-
the d e f i n i t i o n
fl(0'z2) yields
of
=
(since
principle
shows }z21
returning
that
z 1
Iz21 ~ 1 and
fl
lal~
the
- 2~0]
fact
that
a01
- al~ 0
2 1 + 7z 2
7 # 0)
al0 Finally,
0 .
If3(0,z2) I 2 ! i - [z21 2
~ 0
7z~
This
=
~ z2 -
f3(0,z2) ~ 0 Recalling
2 cos ~)(c - % )
we have
Ifl(0,z2) I 2 + An a p p l i c a t i o n
i(2~-$)
conclude
f2(Z) the
that
= ye
_
Consequently
i(2~-$)
$)e i ( 2 ~ $
see
) - c(l-~)
$)e
and
b01
On
~ )eBi~
= ( l - ~ ) e i% + 7(2
and
cos
to
al0
1 al0
~ al0
the d e f i n i t i o n fl(Z)
~ zI
of
and
fl
al0
= i.
we h a v e
= 0
yields
80
f
This
concludes Notice
proper, and "
f
had
7.
q !
The
in
[12]
near
n
~n+l
may
only
that
and
center
badly ÷
to
There
the
date are
is
with
CI(~--~) 2 (2,3)
deg
then
P.j -< 3,
f
was
P(Z) q(Z)
f(Z)
is
used
is
a
j = 1,2,3
holomorphic
only
to
on
show
that
form.
on
those
= S n we h a v e
proper
present of
[5],
the
allows
a~
and
fact
of
balls
in
comparing
[6]
and
that
[II].
for
the
of
seemingly
only
second
and
the
on
is
the
works
those
dimension
of
maps
behave
~n
positive
proper
Cn
present
proper
maps
more
for
in
interesting
smooth
assumptions
At
increasing
construction
The
smoothness
mappings contrasts
the
produced
illusory.
certain
~ ~3
If
f e I
at
about
range
be
following.
questions.
results
contrasts ball
the ~2
f
rational
and
[3],
if
desired
impressive. [2],
and
that
Resume
.
-
of
assumption
the
The
proved
mapping
2
0
(z)
proof.
we
rational
deg
~2
the
that
3
that
case "
of
The
maps
contrast
information
maps
on
the
~
shows
guarantees n
holomorphy
on
the
closed
ball
•
Perhaps
the
most
significant
and
n
pertinent duce
question
was
"pathological"
indicate
that
One
can
n ~ ~n+l
the
proper C1
also
have
posed
at
the
mappings
condition
pose
the
finite
end of
may
question
volume.
as
For
dV
which
is
Lebesgue
proper
of I
into
n the
to w h i c h
on B2
dM
3
+ ~3 + ~
<
~
case
have
Namely
to
Our n+l" case.
proper
the
afj [afj]
we
~4
5,
critical
Considering
measure
mappings
Section
~
be
dM j212][3 fj21 where
of
maps
pro-
results
from
n = 2
and
2
the
following
question.
is 9
2 It
is
known
that
there
exist
f
functions
i
(Z)
on
the
ball
in
Cn
N
which
are
Lip
condition
in
the
ball
algebra
(continuous
on
B
) and
also
satisfy
a
n
~
on
•
Such
functions
can
be
produced
i)
>
n
If l ( Z ) I ~ ~(z
~
1
am
and =
n
s I ]fl(Z)l
=
0,
which
satisfy
8]
where
D
be
a
is
if
is
Lebesgue
coordinate g(Z)
=
n => ~n+l' theoretic
in
measure
a proper
on
S.
map
from
(gl(z),..°,gn+l(Z))
which and
also
U. (P)
=
(Z
such
into
a Lip
functions
~n+l"
a proper
properties
S
~
~n
is
satisfies
topological
Clearly
condition,
can
the
The
continuous
1 [ . gj(Z) I . = P
<
question
measure
sets
0
not
mapping
what
level
can
of
P
gj
<
1
have? As the
a
final
reader
significant (3.3)
is
n
< 2n
< k
that
f even
in
class 2n-i
(2k-l)
these
are
the
these
on
proper
is
some
particular
of (by
examples
of
for
n
were
since also
unfolding umbrella). topics in
it
there
In
aware (and
would
singularity
proper
mappings.
be
space of
< 2n
no
the in
J.
a "planar"
f
is
to
(3.1).
in
will
If
see
mapping
I2(n,k)
a and
with
Under
÷
-
2
the
assumption
the may
interesting might
2n-i Cn
it
was
not
and
large.
The
k
and and
2k-i
=
C 2n-I
pointed Wall)
singularity
manifolds
out
that
4n-3 In
to
one
some
theory.
of
In
mappings
0 .... ,0)
stable and
models
for
suggested
example not
be
to
know
be
C.T.C.
in
is by
smooth
in
occur
(3.3)
of
k =
seminar
the
f
balls
(z 1 .... ,Zn
other
there
if
a paper
with
Damon
mappings
example
theory
following.
they
is
• + B k n T h e r e is
us.
material
stabilizing)
Although
the
(3.3)
singularities
complex
proper
of
from
to
Professors
are
if
equivalent
dimensions
< k
(3.1)
discusses
the
mention
and
~ F a r a n [4] has proven this equivalence. n smoothness assumptions one can see differences
(Zl,...,Zn)
occur
is
mappings
which
real
authors
is
we
(3.1)
thst
interesting
space. the
That
f
certain
[13]
presenting of
then
of
point
example
conjectured
analytic
the
in
2
with
number
Whitney
We
-
is
H.
peripheral
compare
difference.
not.
Thus
and
will
given
a deep if
useful
that by
it
indices. arises
Whitney
connection
other to
these
examples
obtain
more
(the
They in
Whitney
between
the
occuring examples
of
82
References
[1]
C i m a , J. , K r a n t z , S.G. and S u f f r i d g e , T. : A r e f l e c t i o n for p r o p e r h o l o m o r p h i c mappings of s t r o n g l y pseudoconvex and a p p l i c a t i o n s . H a t h . Zeit. 186, 1-8 ( 1 9 8 4 ) .
[2]
C i m a , J. t i o n s to (1983).
[3]
Faran, taking
[4]
F a r a n , J. J. : T h e l i n e a r i t y b a l l s in the l o w c o d i m e n s i o n
[5]
Forstneric, F. : E m b e d d i n g strictly pseudoeonvex balls. T . A . M . S . , 295, 3 4 7 - 3 6 7 (1986).
domains
[6]
Globevnik, J.: Boundary maps". Preprint.
holomorphic
[7]
Globevnik, J. a n d morphic maps from
[8]
H a k i m , M. a n d S i b o n y , la b o u l e u n i t e de C n.
[9]
Lempert, A m e r . J.
[i0]
Lewy, Acad.
[ii]
Low, E. : E m b e d d i n g s pseudoconvex domains
[12]
Webster, S. : On m a p p i n g an n - b a l l i n t o an ( n + l ) - b a l l plex space~ Pacific J. }{ath. 81, 2 6 7 - 2 7 2 (1979).
[13]
Whitney, H. : Singularities space. Annals of ~lath. 45,
principle domains
and S u f f r i d g e , T. : A r e f l e c t i o n principle with applicaproper holomorphic mappings. ~lath. Ann. 265, 1 8 9 - 5 0 0
J. : M a p s l i n e s to
f r o m the t w o - b a l l to the t h r e e - b a l l and m a p s plane curves. Invent. M a t h . 68 4 4 1 - 4 7 5 (1982). of p r o p e r h o l o m o r p h i c case. Preprint.
interpolation
by
S t o u t , E.L. : Boundary the d i s c to the b a l l . N. : F o n c t i o n s Invent. Math.
proper
regularity Preprint.
holomorphes 67, 2 1 3 - 2 2 2
L. : Imbedding strictly pseudoconvex M a t h . 104, 9 0 1 - 9 0 4 (1982).
H.: Naz.
On the Lincei
boundary 35, 1-8
behavior (1977).
of
maps
for
bornees (1982).
domains
holomorphic
between
into
holosur
into
balls.
mappings.
and p r o p e r h o l n m o r p h i c m a p s of s t r i c t l y into polydiscs and b a l l s . Preprint.
of s m o o t h k m a n i f o l d s 220-247 (1944).
in
in
com-
(2k-l)
Finite-Type Conditions
for Real Hypersurfaces
in
En
* John P. D'Angelo University of Illinois Urbana~ Illinois 61801 Introduction Complex function theory in several variables requires a thorough study of the influence of the geometry of the boundary of a domain on the domain itself. us suppose that
~
is an open domain in
real submanifold of information on .
M
~n .
~n
and that its boundary
M
and uses this to derive consequences
M
is a smooth
One measures algebraic-geometric or differential-geometric for the function theory on
In this paper we organize and survey those geometric conditions
crucial role in case
Let
that play a
has a degenerate Levi form.
There are several distinct concepts
that go by the name "point of finite type".
These have arisen since 1972~ when Kohn IKI] first defined the concept for points on the boundaries of smoothly bounded pseudoconvex domains in
~2 .
He established
that this notion was a sufficient condition for subelliptic estimates in the ~-Neumann problem.
In 1974~ Greiner [G] established the necessity of this con-
dition for the estimates to higher dimensions. possibilities.
in this case.
There are many conceivable generalizations
In this paper~ we fit into one framework most of these
We say that
p
is a point of finite 1-type if the order of contact
of all complex analytic varieties with
M
at
p
is bounded.
Catlin [CI~C2] has
proved that this condition is necessary and sufficient for the subelliptic estimates on
(0~i)
forms~ in the case where
More generally~ q-type.
~
is a smoothly bounded pseudoconvex domain.
we study in section I the notions of finite q-type and finite regular
These involve the orders of contact of
complex analytic varieties and
q
then becomes clear that one wishes
q
(perhaps singular)
dimensional complex manifolds~
respectively.
to assign numerical invariants~
between I and n-I ~ that describe the geometry of bounded pseudoconvex domain~
dimensional
M .
In case that
~
is a smoothly
and each boundary point is a point of finite type~
follows from the subelliptic estimates and the work of Bell [Be i] will actually be biholomorphic
invariants of the domain itself.
It
for each integer
it
that these invariants
This is because of
Bell's result that subelliptic estimates imply that a biholomorphism of such domains extends to be a diffeomorphism of the boundaries. We consider here numerical containing
p ~ where
q
invariants
S (M~p) of a real hypersurface q is an integer between i and n-i . We want these
M
* Partially supported by the NSF Grants MCS-8108814 The Institute for Advanced Study.
(A04) and DMS-g501008 and by
84
invariants
to be intrinsic
changes.
to
M
We also want them to m e a s u r %
degeneracy of the Levi form.
at least in the pseudoconvex
Sq
q
dimensional
distinct notions.
cas%
of
complex analytic
submanifolds
of
~n .
These are
semi-continuity
properties.
discuss
the relationship
multitype.
(I~0)
tangent vector fields.
of these ideas to subelliptic
[BJT L
estimates
that arises in the analyticity of
definition
In Section IV we and Catlin's
In section V we consider another notion of finiteness~
Treves-Jacobowitz
for the
In section III
q we consider some choices that arise from generalizing Kohn's original of
q
~n ~ or to be the maximum order of
In section II we consider intersection multiplicities
These numbers have important
using iterated commutators
the
to be the maximum order of contact of
complex analytic subvarieties
tangency of
coordinate
There are many choices.
In section I we consider dimensional
S
and independent of local biholomorphic
CR
due to Baouendi-
mappings.
The
final section contains a list of open questions. This paper has many examples on intersection multiplicities material
is new.
invariants
and references
and some proofs.
in section II and the general organization of the
It seems to the author that the idea of assigning only numerical
to the boundary is too naive; one should assign objects
of ideals of holomorphic
functions
to each boundary point.
algebraic
geometry will be useful in attempting
variants~
a problem beyond the scope of this article.
The author acknowledges partial differential
the participants
equations
thanks the organizers~
of the international
in complex analysis held in
hospitality
such an article.
of The Institute
Finally~
the methods of list of in-
conference
Albany~
1986.
Michael Range and Steve Krantz~
in preparing
such as families
Perhaps
to give a complete
the complex analysis week held at Penn State in March~
encouragement
The results
1985~
on
and of
He particularly
respectively~
for their
he also acknowledges
the
for Advanced Study~ where he wrote this article.
I. Points of finite q-type and finite regular q-type Kohn first introduced real hypersurface
the notion of point of finite type on a pseudoconvex
in the space of two complex variables.
He [KI]
tablish the sufficiency of this condition for subelliptic problem.
When Greiner
estimates~
[G I]
established
the
necessity
was able to es-
estimates
in the ~-Neumann
of this condition
for the
it became clear that the notion of point of finite type was of basic im-
portance in the theory of functions to higher dimensions
of several complex variables.
is not obvious.
priate for several different problems.
In fact~
The generalization
several different concepts
Many different definitions
are appro-
appear in the
literature. In this section we describe how many of these definitions geometric
framework.
It turns out that one describes
fit into one algebraic-
the geometry of a real hyper-
85
surface
M
in
contact
M .
~n
by analyzing how closely ambient complex analytic varieties
Many of the interesting phenomena arise because of the necessity of
considering singular varieties. variables~ Let
(M~p)
denote the germ at
denote the germ of J(M)
This difficulty does not arise in two complex
and thus all the notions turn out to be equivalent in this case.
and
~V)
p
p
of a real hypersurface in
of a complex analytic subvariety of
(V~p)
is the germ of an analytic subvariety.
Let
(V~p)
We will write
for the ideals of germs of functions vanishing on
where the relevant rings are the smooth and holomorphic germs~ Suppose that
~n .
~n .
M
and
V
respectively.
We can always find
a non-constant germ of a holomorphic map I.
z : ~,0) - - >
(V,p)
because any irreducible one dimensional subvariety of measure the contact of
V
the order of vanishing. of
with
M
Since
V
at
V
has a normalization.
p ~ we pull back to such curves~
can be singular~
To
and measure
we must divide by the multiplicity
z ~ the order of the singularity of this one dimensional branch.
This leads to
the notion of point of finite type (which we will call "point of finite 1-type"). Before making the definition; multiplicity of the map
we need some notation.
z ~ and
v(z r)
back map given by composition.
Here
2. Definition.
p
[D i] .
A point
r
We let
denote the
denote the order of vanishing of the pullis any smooth real valued function.
on a real hypersurface
finite 1-type if there is a constant
v(z)
C
M
is called a point of
so that
v(z r)/v(z) ~ C , whenever
z
function for
is a non-constant holomorphic germ as in i; and M .
The infimum of all such constants
p , and is denoted by
&(M~p)
or
&l(M~p)
.
C
r
is a defining
is called the 1-type of
It is easy to verify that this condition
is independent of the defining function. Catlin [C 3] has proved that finite 1-type is necessary and sufficient for a subelliptic estimate on
(0~i)
bounded pseudoconvex domain.
forms~
in case
M
is the boundary of a smoothly
Before proceeding to the more general framework~ we
state a simple proposition. 3. Proposition.
&(M~p)
sup sup [a 6 ~ + :
can also be expressed as lim (dist(z~M)/Iz-pl a exists]
.
Here the first supremum is taken over all one dimensional complex analytic varieties~ the second is taken over in
a ~ and the limit is taken as
z
tend to
p
while lying
V . The function
A(M~p)
locally bounded [DI~2].
is not semi-continuous
from either side~ although it is
To prove the local boundedness
amount of algebraic geometry~
seems to require a certain
especially the notion of intersection multiplicity.
86
This is the motivation for section II of this paper.
Before turning to these ideas~
we begin a general discussion of finite type conditions~ we would like numerical invariants of number of possible definitions
M
to have.
by listing the properties
We then proceed to a large
that satisfy only some of the properties.
Only the
multiplicities of section II will satisfy all the properties 4 (including upper semi-continuity)
below.
4. Desired properties of numerical invariants. hypersurface integer
q
in
~n .
between one and
n-I ~ to the germ
4.0
S (M~p) q
4.1
Sn_l(M,p) J Sn_2(M,p) J ... ~ SI(M~p)
4.2
Let
(M~p)
be the germ of a real
We wish to assign numerical invariants (M~p)
Sq(M,p) , for each
that satisfy the following:
is a positive real number or plus infinity.
Sq(M~p) = 2
.
if the Levi form has (at least) n-q eigenvalues of the
same sign at
p .
In particular~
all the numbers equal 2 when
strongly pseudoconvex from one side at
p .
M
is
(See section 3 for the
definition of the Levi form.) 4.3
S (M~p) is an invariant; this number does not depend on a choice of q local coordinates or on a local defining equation for M .
4.4
S (M~p) is a locally bounded function of p . q an upper semi-continuous function of p .)
4.5
S (M,p) is finitely determined. Suppose that M is defined by r q in J ( M ) and that S (M~p) is finite. Then there is an integer k q so that Sq(M~p) = Sq(M',p) whenever M' is defined by any r' that has the same
Finally~
k
jet as
r
at
p .
one hopes that these numbers have an intuitive geometric definition and
that they arise in several applications. 4.4 and 4.5.
at
p :
Let
Q
We recall a definition from
smooth germs~
Let J
be an ideal in
where the infimum is taken over constant holomorphic maps 6. Definition. of
q
z
The invariant
g
in ~
~Q .
z's
and the
We put
formal power series
~'s .
T (J)
= sup inf v(z*g)/v(z)
and the supremum is taken over all non-
as in definition i. &q(M~p) ~ a measure of the maximum order of contact
dimensional varieties with q
in a slightly more general
real analytic germs~
z's ~ or formal power series in the
5. Definition.
[D i],
denote any of the following rings of germs of smooth functions
holomorphic germs~
in the
Many of our candidates will not satisfy
In order to obtain these for orders of contact~ we must consider
singular varieties. context.
(Even better would be
M
at
(M~p) = inf T ((~,Wl,...,Wq_l))__
p ~ is defined as follows:
87
Here
J
by J
is the ideal
~M)
~ and the ideal in question
and q-i linear forms at
linear forms.
p .
Note that~ when
7. Definition.
q
The invariant
supremum being
taken over all
q
is the ideal generated
The infimum is taken over all such choices of
equals one~
the definitions
2 and 6 are equivalent.
T (M~p) is defined as in Proposition 3~ with the q dimensional complex analytic varieties. The in-
variant
®q(M~p)
is defined as in Proposition
3, with the supremum being
over
dimensional
complex analytic manifolds.
Alternative
q
~regq(M~p)
are
8. Remarks.
reg Oq(M~p)
in [K2]
It is easy to verify
satisfy properties these numbers
4.1~ 4.2~
and
aq(M~p)
and 4.3.
we see that when
q
equals
See Example
n-i ~ all the invariants
gregq(M~p)
We say that
is f i n i t %
p
in
and that
p
of definitions
It is also clear that
also explains w h y there is only one viable concept 9. Definitions.
M
taken
for
in [B I] .
that all the invariants
are not finite simultaneously.
notations
6 and 7
® ~ T .
I0 below.
However~
In Section III
give the same values.
This
in two complex variables.
is a point of finite regular q-type if
is a point of finite q-type if
~q(M~p)
is
finite. i0. Example.
Let
r z,7) Let
p
i0.I
= 2m ; gmven by = =
®2(M~p ) = 6 ;
given by
= ~3(M,p)
ii. Proposition. Proof.
Let
N
Ii.i
dist(z~M)
Whenever
z
T (M~p) q variety V
J const
image lies in
V'
for the linear
imbedding
Letting
t
.
z4 = 0 .
.
iterated
cor~nutators
of the function
&
q
in Section III.
(M~p)
.
. From its definition we can find,
for any positive
for which
I z-pla p
the variety
least one dimensional.
o
.
given by
some of the properties
is close to
linear forms~
z I = z4 = 0
= 4 ;
denote
values:
2 3 z I - z 2 = z4 = 0 .
T q (M~p) -< & q (M~p)
a q-dimensional
.
2 3 z I - z 2 = z 3 = z4 = 0
to this example when we discuss
We now describe
m~4
z I = z2 = z 2 = 0
; given by
= 2m ; given by
greg3(M~p)
;
We have the following
TI(M,p ) = &l(M~p)
We will return
$ #
2 + Iz3r2m
-
denote the origin.
&2(M,p) 10.3
be defined by the equation
2Re z4) +
=
®l(M~p)
10.2
M
and lies in V'
defined by
We can therefore
The inequality
tend to zero~
from
V ~ and J(V)
and writing
If we choose
z
as in Proposition
along the image of
z .
i whose
Write
into E n given by these linear forms. ~
for
q-i
and these linear forms must be at
find a map
ii.i holds
~n-of+l
N-E < a < N .
z w ~ we obtain
that
w
88
Ir(~(t))l
J const Itl av(~) ,
and hence that
v(~ r)/v(~) > a > N-~ .
12. Theorem [D i] .
Let
M
This implies the desired result.
be a smooth real hypersurface
of points of finite q-type is an open subset of
M .
of
~n .
In fact~
Then the set
the function
~q(M~p)
is locally bounded. 13. Theorem.
Let
M
be a real analytic hypersurface
of
~n .
Then
p
of finite q-type if and only if there is no germ of a complex analytic subvariety of
~n
containing
14. Proposition.
p
and lying in
The function
&q(M,p)
is a point q
dimensional
M .
is finitely determined
in the sense of
property 4.5. Proof.
If
&q(M~p)
Suppose that
r'
is f i n i t %
14.1
k
larger than
is any smooth function with the same k-jet as
is any defining function for for a generic
choose any integer
w .
M
.
For such a
C=~q(M~p)
r
at
.
p ~ where
r
The infimum in definition 6 is clearly attained
w ~ and any nonconstant holomorphie
z ~ we have
z w r' = z w r + z w (r-r')
The second term on the right of 14.1 vanishes the first vanishes
to order at most
kv(z)
to order at least
by the choice of
(k+l)v(z)
k .
~ while
Therefor%
left side vanishes also to the same order as the first term on the right.
the
Hence
v(z w r')/v(z) J C ~ so property 4.5 is satisfied. 15. Corollary.
The function
~q(M~p)
satisfies
all the properties
4~ although
it
fails to be upper semi-continuous. Proof.
The failure of semi-continuity
4.5 in proposition
can be seen in example II.16.
14~ and stated 4.3 in Theorem 12.
trivially from the definition of
We have proved
The property 4.1 follows
A (M~p) , and the properties q
4.2 and 4.3 are safely
left to the reader. If we write
Mk
for the hypersurface
see from Proposition bilizes
to
&q(M~p)
16. Example.
Put
Suppose that
m ~ 7
£1(Mk, P) = =
defined by the k-jet of
14 that~in the finite q-type c a s % .
~reg
at
p ~ we sta-
r(z) = 2Re(z3) + I z~- z~12 + i Zl18- I z2112 + I zll2m and that
p
is the origin.
for
0 < k < 7
= 12
for
8 < k < ii
= =
for
12 < k < 2m-i
= 3m
for
2m < k <
(Mjp)
r
eventually
Here is a simple example from [DI].
We complete this section by noting that properties numbers
~q(Mk~P)
The 6 jet of example
q both of these properties
fail when
q
Then
4.4 and 4.5 fail to hold for the
16 furnishes us with an example where
equals one.
89
II. Multiplicities There are many possible ways to measure a singularity; intersection multiplicities are one of the nicest. hypersurface.
In this section we show how to define such numbers on a real
This yields a collection of numbers
B (M,p) that satisfy all the q Before proceeding to the necessary algebra~ the following
properties of Section I.
example compares these numbers to the ones we have considered thus far. Example i.
Put
suppose that
r(z~-z) = 2Re(zn) +
n-I 2m. ~ Izjl J ;
let
p
denote the origin, and
m I _> m 2 > ... > mn_ I .
(I.i) A (M,p) = Areg(M,p) = 2m q q q n-i (1.2) Bq(M~p) = 2 - ~ m. q J In example i~ the collections of numbers (i.i) and (1.2) convey the same information.
However~
the number
BI(M~p )
includes all the information.
number that the author feels is the most useful.
It is this
To define it~ we need some basic
formal algebraic notions. Notation 2.
Let
p
be a point in
~n
We consider several local rings at
~denotes
the holomorphic germs~ d d e n o t e s
~denotes
the formal power series in
/
power series in
(z-p ~ z-p) .
z-p ~ and d
We say that
p
the real analytic, real valued germs~ \
denotes the formally real formal
~7 Cab(Z-p)a(z---~)b
is formally real if
Cab = Cba Note that the Taylor series of a defining function for a smooth real hypersurface /
containing
p
k
gives us an element of ~ ' .
this in terms of
~
.
result will be in terms of ~ Computation 3.
Put
Cab(P ) = Cba(P ) .
Our next computation shows how to write
In case the defining function is an element of ~
the
.
w = • Cab(P)(z-p)a(~-p)b
We d e f i n e elements of ~
," assume that
Co0(P) = 0
and that
as f o l l o w s :
(3.1) hP(z) = 4 ~ Cao(Z-p)a (3.2) fP(z) = (z-p) b + ~ C a b ( P ) ( z - p ) a (3.3) gP(z) = (z-p) b - ~ Cab(P)(z-p)a Then we have (3.4) 4w = 2Re(hP(z)) + Note that~ if that c a s %
w
llfP(z)112 lies in ~
HgP(z) II2 hP ~ fP ~ gP
are all elements of
according to [D i~ D 3] ~ the only complex subvarieties of
in the zero set of
w
~n
must be defined by the equations (4.1) and (4.2):
~
.
In
that can lie
90
(4.1)
hp = 0
(4.2)
fP = u g P
Here
U
f~ = ~ Ubkg P k
or
is a unitary matrix of constants. This motivates the definition of the ideals
~d~(U,p)
in ~ d e f i n e d
Definition 5.
by
~(U,p)
is a proper ideal in ~
is generated by
hp
(or ~
w
in case
and
f~
-
~ U b kg ~ "
Note that J ( U , p )
is in ~ ) .
These ideals are the obstructions to finding complex analytic subvarieties in a real hypersurface.
In other words, we have the following restatement of a result
from [D i]. Theorem 6. w
Let
be a d e f i n i n g
M
be a real analytic hypersurface of
function
for
M ~ and l e t
~(U,p)
dim V ( ~ ( U , p ) )
< q
~n
containing
be t h e
ideals
p
Let
of definition
5.
Then
(6.1) ~q(M,p)is finite < = > Now~ in case
dim V ( ~ ( U , p ) )
equals
0
(for all
for all
U )
U , there is a simple way
to measure its singularity,
We recall some analytic geometry [S,D i]. /k Definition 7. Let J be a proper ideal in ~ or ~ . Its multiplicity, /k is the dimension of the complex vector space ~/~ , or ~/~ Theorem 8.(Nullstellensatz).
Let ~d~ be a proper ideal in
~
.
D(~F),
The following are
equivalent. (8.1) D ( J )
<"
(8.2) V(.j ~) = [p]
(or,
dim V ( J )
(8.3) There is an integer vanishes at Remark 9.
p
(s
In case J
s
: 0 ).
so that
~s
lies in J
is independent of
for every
~
that
~ )
is defined by specific generators whose coefficients depend
continuously on some parameter~
then
D ( ~ d~)
depends upper semi-continuously on this
parameter. Example I0.
n
Suppose
f :(~ ,0)
> (~n, 0)
ideal generated by the components of of
f .
f ~ i.e., its winding number about O.
generic number of roots to the equation coefficients of
f
For example, if
J=
is holomorphi% Then
and J
denotes the
is the topological degree
It also has the interpretation as the f(z) = w
depend on some parameter (ez+ z2) , we have
D(J)
for
w
close to
s ~ we have that I = D(4)
0 .
D(~)
If the
~ D(,~0)
j D(~0 ) = 2 .
We extend the notion of multiplicities to varieties of positive dimension as follows. plane
Suppose that
P , the point
dim(V(~&~)) = q . p
Then~ for a generic choice of an
is an isolated point of the intersection of
V(J)
n-q and
P .
91
This follows from the local parameterization Definition (ii.i)
ii.
Let J b e
D q (J)
theorem [W].
a proper ideal in
= inf D ( ~
~
Lemma 12.
.
We define
q-i
linear forms.
< ~ <
q
>
dim V ( J )
< q .
This is immediate from the remark before definition
We now return to our real hypersurface define from it ideals
J(U~p)
.
13.
Theorem 14.
M . Choose a defining function,
and
function.
Bq(M~p) = 2 SUPu D q ( J ( U , p ) ) Aq(M,p) _< Bq(M~p) __< 2(Aq(M~p)) n-q .
are simultaneously Proof.
ii.
We will sketch a proof below that the result will
be independent of the choice of defining Definition
We adjoint these
and take the multiplicity.
D (J)
Proof.
~
I ..... ~q_l ) ,
where the infimum is taken over all choices of to the ideal~
or
We have the
In particular,
the two numbers
finite.
The proof is similar to the proof of Theorem I.i0.
In case
q
equals one
this is proved on page 630 of [DI].
The point is that by using the ideals f ( U , p ) *( * Aq(M~p) _.< 2 sup J(U~p)) , where T is the invariant deU defined by pulling back to one-dimensional varieties as in Proposition 1.7. This
one estimates
invariant
that
is smaller than or equal to the (smallest)
integer
which works in 8.3,
which in turn is smaller than or equal to
D(J(U,p))
in
comes from estimating
~ / J
(U,p))
terms of one.
.
T (J(U~p))
However,
Theorem 15. Proof. (15.i)
as in
[D i].
q-i
Given
q
is upper semi-continuous
P0 ~ we must show that if
ii.i).
p
as a function of is close to
equals is
This amounts
space. p .
PO ~ then
there is nothing
to prove.
Suppose that it is
We have BN (M~p) = 2 sup U inf D ( ~ ( U , p ) , ~
it follows from formulas
the generators of , f ( U ~ p ) D
q
is easy because the q-multiplicity
in
B q (M,p) _< Bq(M, p0 ) .
finite.
Now~
D(~d~(U,p))
equals one in a lower dimensional
In case the right side is infinite
(15.2)
q
linear forms to the ideal (Definition
the case when Bq(M,p)
(One lists the monomials
This is a sketch of the proof when
to prove this for higher
defined by adjoining to considering
The second inequality
.
s
3.1,
1 ..... ~q_l ) •
3.2~ 3.3~ 4.1 and 4.2 that the coefficients
are continuous
functions of
is usc. as a function of these parameters.
functions
is usc., we have
close to
P0
Dq(~(U~p))
U
and
Since the infimum
is usc..
p .
of a family of usc.
This function is finite for
because of theorem 14 and theorem 1.10.
of
By remark 9~
p
(The proof of the latter fact
92
uses semi-continuity
when
q
equals one.
it has only finitely many values, semi-continuity.
See [D i].)
Since it is integer valued~
so taking the supremum also preserves
the upper
Hence 15. I holds.
p = origin. Then Example 16. Put r(z,-z) = 2Re(z3) + I z ~ - Z e Z 3 1 2 + [z214 2 2 2 BI(M,p) = 2D(z3, zel-ZeZ3, Ze) = 2D(z3, zl, z 2) = 8 . According to theorem 15, BI(M~p) < 8 fact that
for all
p
close to the origin.
Al(M, o r i g i n ) =
4 ~ but that
This is to be contrasted with the
&l(M,p) = 8
for certain
p
arbitrarily
close to the origin [D 2]. Lemma 17. Proof.
Bq(M,p)
does not depend on the choice of defining equation.
We sketch the proof.
Let
r
and
r'
~ sup D q ( J ' ( U , p ) ) U
,
be defining functions.
It is
enough to show that (17.1)
sup D q ( J ( U , p ) ) U
because we can then interchange simplicity we assume that formally as in 4, and
q
the roles of
equals one.
r' = 2Re(H) +
r
and
We write
r'
to obtain equality.
For
r = eRe(h)+ II f II2 -llg I]2
IIF II2 - fiG 112
We assume that
r' = ur
where (17.2)
u = i +llall 2 -llb 112
Note that (17.2)
is not in the same form as (3.4),
the pure terms.
This is because
So we work modulo function in
this ideal.
a ,
a copy of
the ideals
(h)
because
there /is % no need to isolate (H) in ~ are equal [D4].
and
Given any U , we choose a copy of U for each * U for each function in b , and consider the unitary
matrix
(17.3)
Using
U' =
0
U
0
0
0 -U*
F = f @ af @ bg G = g@
Computing
ag@
F -U'G F-U'G
Hence the ideals (17.6)
0
0
(17.2) we obtain that, with evident notation~
(17.4)
(17.5)
U
bf
by using
= (f-Ug, F -U'G
(17.4) and (17.3)~ we obtain
a ( f - U g ) , BU ( f - U g ) ) and
f -Ug
are the same.
sup D ( f - U g ~ h) ~ sup D(F -UG,H) U U
This is (17.1) in case
q
equals one.
.
.
Therefore we must conclude
that
93
Remark 18.
It follows from (8.3) that
Bq(M~p)
is finitely determined,
this invariant satisfies all the properties of Section I.
so that
It would be interesting
to give an intrinsic formulation of this number. Example 19.
r(z)=
2Re(z4)+
Iz21 - z2z312 + Iz~-z53 12 + Iz7312 .
5 7 Bl(M, origin) = 2D(z4, z2I- zmZB, Z~- z3, z3) = 84 .
5 7 B2(M, origin ) = 2D(z4, z21 - z2z3, z~- ZB, Z3, Z3) = 12 . 2 3 5 7 B3(M, origin) = 2D(z4, z I- z2z3~z 2 - z3, z3, z3, z2) = 4 . Al(M, origin) = 14
;
given by
(t4~t5~t3,0)
A2(M, o r i g i n ) =
6
;
given by
(st, s2, t2,0) .
A3(M~origin) =
4
;
given by
(s,t~u,O) .
III. Iterated c o m u t a t o r s Kohn's [KI]
.
and related conditions
original definition of point of finite type in
iterated cormmutators.
~2
involves
Because of the importance of this technique in partial differ-
ential equations~ the notion has many applications.
We recall the generalization,
to Bloom-Graham [BG], of Kohn's original definition.
due
We also discuss related con-
ditions on vector fields due to Bloom [B I]. Since manifold. TIOM
M
is a real hypersurface of
Let
TM @ ~
~n ~ it inherits the structure of a CR
denote the complexified tangent bundle to
denote the integrable subbundle of
fields tangent to The Levi form
M .
X
TM ~ ~
We write, as usual,
whose sections are (i~0) vector
T0~
measures the extent to which
for the complex conjugate bundle. TIOM @ T 0 ~
To define it, choose a nonvanishing real 1-form let (i)
L
and
L'
k(e,e') =
be local sections of =
<~
TIOM . [L,~'] >
M , and let
~
fails to be integrable.
annihilating this direct sum, and
We put .
We now define the type of a vector field at a point. Definition 2. the type of
Let L
at
L
be a local section of
p ~ written type
there is an iterated commutator (2.1)
<~, X>(p)
L(p) ~ 0 .
(L,p) , to be the smallest integer of
L
and
~
of length
(m-l)
We define m
so that
so that
# 0 .
By a commutator of length (2.2)
X
TIOM ~ with
(m-l) , we mean a vector field
X = [... [[AI, A2],A 3] "'" A ]
where each
A. is either L or ~ . ] One observes that the type (L~p)
X(L~L)(p) • does not vanish at
equals two precisely when the Levi form,
p . More generally, we can define the types of
94
subbundles.
Let ~
be a subbundle o~
TIOM .
We let ~ m ~ )
denote the module
over the smooth functions generated by commutators of local sections o f ~ their complex conjugates of length
m-i .
Definition 3. The type of the s u b b u n d l e ~ a t smallest integer (3.1)
m
and
written t y p e ( ~ , p )
p
for which there is an element
X
of
~m(~)
~ is the for which
< ~ ~ X > (p) ~ 0 . Following Bloom~ we can now give another family of notions of finite type.
Definition 4 [B i].
tq(M~p) = s u p [ t y p e ( ~ , p ) ]
all subbundles of
TIOM
of dimension
~
where the supremum is taken over
q .
This concept has no relationship to the intersection theory of
M
with complex
analytic varieties or manifolds in general, although it seems to when
M
convex.
equals
Not much is known about the numbers
tq(M,p)
except when
q
is pseudon-i .
In that case we have Theorem 5.
Let
M
be a smooth real hypersurface of
~n .
All of the following
numbers are equal: (5.1) tn_l (M~ p) (5.2) ~ e ~ ( M ~ p ) (5.3) ~n_l (M, p) (5.4) ~ n _ l ( M , p) (5.5) ~ n _ l ~ l ~ p )
(see definition 111.7)
(5.6)
(see definition IV.7 and theorem IV.10.2)
m2(P)
Remarks on the proof.
That (5.1) equals (5.2) is proved in [BG].
(5.3) are equal is proved in [D i] [B 1 ].
and [K2].
To include (5.4)~ note that the process of computing
computing the order of vanishing of the defining equation where restricted to Unfortunately~
That (5.2) and
That (5.5) equals (5.1) is proved in Bn_ 1
amounts to
2 Re(z n) + higher terms~
z =0 . n the condition that tl(M~p) be finite has a serious disadvantage.
It is not an open condition~ and if assumed locally, it is not finitely determined. Example 6.
Put
r ( z ) = 2 R e ( z 3) + Iz~- z$12 + f(zl~z2) ;
Suppose first that that type
(L~p)
~/~z I
L
in
L
vanishes identically.
vanishes or not.
tangent to
p
is the origin.
Then it is elementary to verify
equals 6 or 4 ~ depending only on whether the coefficient of
hand~ along the variety for
f
where
V
V and
We have
defined by ~
is finite is not an open set.
in
t2(M,p) = 4 , tl(M~p) = 6 . On the other 2 3 0 = z 3= z I - z 2 ~ we have type ( L ~ ) = =
V ~ not the origin.
Thus the set where
Worse yet~ suppose now that
f
tl(M~p)
vanishes to infinite
95
order at
p , but is strongly plurisubharmonic as a function of its two variables
elsewhere. type
Then we have type
(L,p)
(L,p)
equals 2, for all
equals 6 or 4 as before.
Thus type
L , unless
(L,p) < =
p=p
for all
, whereas
L
and
p
near the origin, so the condition (assumed locally) is not finitely determined. There is a related invariant introduced by Bloom [B i] and used also by Talhoui IT] (%~))
Suppose first that ~
is a subbundle of
TIOM
and that trace
denotes the trace of the restriction of the Levi form to . ~
C (~,
p)
fields,
denote the smallest integer
m
for which we can find
LI,...,Lm_ 2 , that are local sections of ~ o r
.
m-2
Let
vector
complex conjugates thereof,
for which (6)
(Lie2"..Lm_2)(trece k ~ ) )
(p) # 0
Definition 7.
C (M,p) = s u p [ C ( ~ , p ) ] q dimensional subbundles . ~ of TIOM .
~
where the supremum is taken over all
q
The numbers of definition 7 satisfy properties (4.1) through (4.3) of section I, but not the last two.
It is also easy to verify that
(7.1) &~eg(M,p) _< Cq(M,p) To illuminate the definition, vector field
L .
Then we have
iterated commutators of and
~ .
verse to
L
suppose that ~
trace k(O~)
and
.
Thus~ instead of taking
~ , we differentiate the Levi form applied to
The two notions are not identical. T I0 @ T Ol
is the bundle generated by one
= k(L,L)
for which
Let
= I .
T
L
denote a vector field trans-
Then we have, with evident notation~
(8.1) [[L,E],L] = [k(L,L)T + AI0-~I0, L ] The contraction of (8.1) with
~
gives
(8.2) -LX(L,L) + ~L'X(L,L) + X(L, AI0) for some function
~L ' not in general zero.
Thus, even up to order three, there
are terms in the iterated commutator that do not appear in a derivative of the Levi form.
Note, however, that if
must vanish if type
(L,p)
and
k(L,L)
does.
C~,p)
,
k
is assumed semi-definit%
(~=
bundle generated by
conjectures even more; namely that~ if Conjecture. 8.
then the term
(M pseudoconvex)
l
L ) are equal.
is semi-definite,
Example 9 [B i].
Put
Bloom [B i]
then
Cq(M,p) = tq(M,p) = Areg(M,p)q .
We now turn to an interesting example in the case where
be the origin.
X(L, AI0)
In that case one expects that the two invariants,
r(z) =
2Re(z3) + I Z l I 4 + 21Z112 +
g ~z 3
,
M
is no___~tpseudoconvex.
2(z2+E2)IZ112 ,"
let
p
Let
__~+ L = ~i'~I where
g
~z 2
is chosen so
L
will be tangent to
M .
It is elementary to verify that
96
[[L,~]~L]
vanishes
identically.
(9.1)
tl(M,p ) = CI(M~p ) = =
(9.2)
Aleg(M,p)
= ~l(M,p)
We have
;
t2(M,p) = C2(M,p) = 2
= 4 ;
A~eg(M,p)
= A2(M,p)
= 2
From this example one sees that these numbers of contact in the general (9.3)
Areg(M,p) q
has proved
[T] proves
conjecture
8 in the special
a weak version of it in
the rank of the Levi
form,
[D5] has proved a related
this result result.
the Levi form is semi-definite
to orders
M
lies in
Because of his assumptions
is not really any more general.
Let
field
L
the number
C~p)
~ where ~
type
.
c _< max(N~2N-6)
Then
~n .
case where
M
be any CR manifold~
~3.
on
The author
and suppose
that
on the span of the (i~0) parts of all of the iterated
cormnutators of a (I~0) vector
(L~p)
are not related
< t (M,p) -- q
Bloom [B i] Talhoui
C and t q q One does always have that
case.
and its complex conjugate.
is the bundle .
generated
Perhaps
by
c
and
N
t
and C
Let
c
L , and let
denote N
denote
are equal under
these
hypotheses. One sees from example 6 that the invariants
q This casts doubt upon their usefulness;
determined.
that they can be defined one could formulate numbers
on any CR manifold~
intrinsically
arise in subelliptic
On the other hand~
scalar partial differential II
Hormander Theorem
i0 (Hormander
0
tru%
play a large role in the theory of
The interested
there.
in
reader should
We mention one well-known
[H]; see also Kohn [K3]).
vector fields defined near
conjecture were
Areg(M~p) . Since none of these q the interest in them has subsided somewhat.
commutators
operators.
[HI and the references
and if Bloom's
the number
estimates~
iterated
are not locally finitely q however they have the advantage
IRn .
Let
Suppose
P
denote
that
consult
example:
L0, LI~...~L N
are
the second order operator
N
P = L0 + Z L2 i j Then~
P
is subelliptic
cormmutators of length
<=>
there is an integer
less than or equal to
m
m
so that the
L's
and their
span the tangent space at
0
in
]Rn The ~-Neumann as in Theorem However~
i0.
problem on (0~n-1)
forms exhibit essentially
In this case the iterated
the finite
~-Neumann problem on
type conditions (0~q)
forms.
commutators
of sections Presumably
find their way into the (as yet not understood)
IV. Subelliptic
estimates
We will not describe However~
if
~
control
the same properties the situation
I and II are required variations
[K2].
in the
of those ideas will
study of subelliptic
systems.
and Catlin's multi-type the ~-Neumann
problem in any detail here.
is a bounded pseudoconvex
See [K2~CI].
domain with smooth boundary~
and
~
is
g7
a
~-elosed
(0~q)
form with
L2
coefficients~
structs a particular solution to the equation know when
u
must be smooth wherever
~
then the
~u=~
is smooth.
.
~-Neumann problem con-
One obviously wishes to
This local regularity
property follows from the subelliptic estimates described below. Suppose that
p
is a point in
bd~ .
subelliptic estimate on
(O~q)
forms at
C~E
U
p
and a neighborhood
of
The
p
D-Neumann problem satisfies a
if there are positive constants
such that
]l + l]2 < C( I]~ 112 +]1~*~ 112 + ]]~ ]]2)
(i)
E
whenever
--
~
is a smooth form~
in the domain of
follows from the work of Kohn that when
bd~
c
is strongly pseudoconvex at
equivalent to the estimate for some
~
~ and supported in
can be no larger than
~
p .
U
.
It
1/2 ; this occurs
The search for geometric conditions
smaller than
1/2
has motivated most of
the work described in this paper. Kohn [K2] invented an iterative process for proving the estimate i; his method gives only a rough approximation
to the actual value of
the ideal of germs of smooth functions
f
at
p
for which i holds~ when the left side is replaced by to find conditions
He considered
]If~ II~
One then wants
that imply that the constant function i lies in the ideal.
There are certain functions~ form~
E .
such that there is some positive
that lie in this ideal.
for example appropriate minor determinants of the Levi By the processes of differentiating~
evaluating
determinants and computing real radicals of an ideal~ Kohn defines ideals iteratively. 2.
3.
He proves that
If I lies in on
(0, q)
If
M=bd~
l~(p)
for some
k ~ then there is a subelliptic estimate
forms. is a real analytic h y p e r s u r f a c %
contains no germ of a complex analytic This relies on the Diederich-Fornaess M
q
then i lies in
theorem [DF].
is the boundary of a bounded domain~
l~(p) < = >
dimensional variety at In particularj
and is real a n a l y t i %
p . if
this must
be true. Combining this with the results of Catlin b e l o %
we have the following
implications: 4.
C~ i
5.
case in
l~(p) = >
subelliptic estimate < = >
gq(M,p) <
Real analytic case I
in
l~(p)
Ik q <=> <=>
subelliptic estimates < - - > M
contains no germ of a
~q(Mzp) < q
complex analytic variety at
dimensional p .
M
98
It remains important
to verify the remaining
ideals can be defined on CR manifolds Catlin's proof of the estimates constructs
plurisubharmonic
(or infinity)
from the properties
~i -< li+l
and give some examples. that
function
i .
7.
Suppose
r .
A weight
that
~(p)
M
a
Thus,
weights Examples (8.1)
> 0
in case
AI
is finite,
with
i _< I.l -< ~ ' and We also
if it is finite by
of
is distinguished
~n , with defining at
p
if~ in some
> D~DBr(p)-- = 0 .
= (ml, m2, ...,mn) direction
z. . i
(in the lexicographical
ordering)
weights. we can think of However~
mi
as a weight to associate
one inductively uses the previous
to define them. 8. r(z) = 2 Re(zn) +
n-I ~ Izj[ i
2pj '
Pl -< P2 -< "'" -< Pn-I
= (l~2Pl ~2p2 ,...,2pn_l)
r(z) = 2 Re(z3) + [z21-mS[ 2 ~(0)
= (1,4,6) --
(8.3)
~i ~
is a real hypersurface
(Xl, k2~ ...,In)
list
We do warn the reader that a finite
are integers.
is the smallest weight
if ~ p )
~(0) (8.2)
They also involve some of the
are ordered lexicographically.
j--
as he
are basic to his proof of the estimates.
is defined,
that is larger than all distinguished
with the coordinate
from Kohn~
In his proof he also
be finite; however,
These weights
coordinate system n (7.1) ~ (~i+Bi)~i I < I The multi-type
respects
an n-tuple of positive rational
A weight is an n-tuple of numbers
demand that i = h I and that I k k (6.1) ~ a.I -I = I ak > 0 J J ' Definition
AI
also is and its properties
for all
as well [K4].
Here we do little more than recall the definition,
does not guarantee
Definition 6.
in 4~ because Kohn's
His numbers are ordered oppositely
I.i through 1.5 of section I.
the basic properties~
the multi-type
He associates
to a boundary point.
concepts of section III.
multi-type
differs in technical
function with large Hessians.
uses the concept of a multi-type. numbers
implication
and have other applications
5
r(z) = 2 Re(z3) + [Zl14 + I z218 + 2 Re(ZlZ2) ~(0)
= (1,4,20/3)
A2eg(M, 0) = 4
;
A2eg(M, 0) = 8
Observation 9. Consider example (8.2). Let p be a point along the variety 2 3 z I= z 2 close to 0 and lying in M . Then the multi-type at p will be (1,2,~) ,
unless
p
is the origin.
Thus there are points close by a given point
99
of finite m u l t i - t y p % sens%
the weight
with infinite multi-type.
(I~2~)
is smaller than
However~
(i~4,6).
in the lexicographical
This sense of upper semi-
continuity holds and is one of the main points in Catlin's work. Theorem i0 (Catlin).
Let
M
be pseudoconvex.
satisfies
the following properties:
i0.i
~
is upper semi-continuous
P0
in
M
is given~
~(P) 10.2
_< ~ ( P o
If ~ ( p 0 have
10.3
m
holomorphic
of
sense; i.e.~ PO
if
on which
~
then
mn+l_ q _< ~q(M,p 0) .
If
q=n-I
, we
.
q .
dimension.
Suppose ~ ( p ) ~(p)
U
is finite. Then the set of points p close to P0 n-q ~ / ~ p ) = ~ ( p 0 ) is contained in a submanifold of holomorphic
dimension at most
10.4
nbd.
)
) = (ml~...,mn)
for which
in the lexicographical
then there is a
m 2 = An_l(M~P0)
Suppose
Then the multi-type ~ ( z )
See [C2] or [KI]
Thus 10.3 gives us a stratification
is finite.
is distinguished
or [DF] for the concept of of
M .
Then after a change of coordinates~
in the sense of 7.1.
The proof of this theorem is quite long and involves an alternative which Catlin calls the commutator multi-type. subellipticity
is the stratification
of this paper and property 10.3 that locally contained
in manifolds
M
dimension q-i .
for
from theorem I.lO
in this sense;
Thus;
the multi-type
except for
is smaller nearby~
"better behaved". The reader should consult [C3] when it appears~
versions of these results Als%
The key property
It follows
is a finite union of sets which are
of holomorphic
points on a set that is negligible hence
See [C2].
given by 10.3.
definition
are required
to establish
Catlin has an alternative measurement
where even more precise the subelliptic
estimates.
of the order of contact of q-dimensional
varieties.
V.
Essential
finiteness
In this section we discuss an important concept due to Baouendi-JacobowitzTreves [BJT]. Theorem i. that
They proved the following Suppose that
~ :~-->
M
~
and
M
important
are real analytic hypersurfaces
is a CR diffeomorphism.
Then~
if
M
is essentially
of
~n
and
finite~
is actually real analytic. We give the definition of essential show its relationship
vanishing of a real analytic the definition
that
finiteness
to the other notions.
M
function
be essentially
r(z~)
and several propositions
We suppose that ~ that
finite at
0 .
0 M
M
lies in
that
is given by the M ~ and we give
is then essentially
100
finite if it is essentially finite at each point. Definition 2. r(z,~) = 0
M
is e s s e n t i a l l y finite at
imply that
z=0
0
if the equations
r(0~)
= 0
and
.
In fact this definition is independent of the choice of d e f i n i n g function.
The
following propositions give some insight into this concept. P r o p o s i t i o n 3. If
&l(M~p)
If
M
is f i n i t %
is e s s e n t i a l l y finite at then
M
p ~ then
is e s s e n t i a l l y finite at
&n_l(M~p) p .
is finite.
Unless
n = 2 ~ the
concept is not e q u i v a l e n t to any of the intermediate notions considered earlier. Proof.
See [BJT]; both are easy.
N o t e that n e i t h e r converse is true unless
n
equals 2. We h a v e seen in II.3.4 that every real analytic function can be w r i t t e n as (4)
2 Re(h) +
In case
g
IIf I12
vanishes~ w e h a v e e q u a l i t y b e t w e e n 1-finite-type and essential finiteness.
P r o p o s i t i o n 5. finite at Proof. =>
Suppose
0 <=> <=
To prove
M
is defined by
2 Re(h) +
IIf II2
Then
M
is e s s e n t i a l l y
~I(M~0) <
this is part of p r o p o s i t i o n 3. that
See theorem 11.6. hand~
llg II2
&I(M~0)
is finite~
Suppose that
the d e f i n i t i o n of essential
r(0,~) = h(~) = 0 ,
then
z
it is enough to show that
z E V(h~f)
.
finiteness
equals
0 .
Then
r(z,~) = ~(~)
tells us that~ Thus
V(h~ f)
if
V(h~f) = [0] .
r(z~)
.
On the other = 0
whenever
is trivial and we are
finished. Finally we give a necessary and s u f f i c i e n t condition that is easy to verify. Let M be defined by r, w h i c h we write, a f t e r a local coordinate change, as (6)
r(z,z)
= 2Re(z n) +
~hb(z)zb
2Re(z n) + Zha (z) (~,)a + ~n S
Here each h b vanishes at 0, each m u l t i - i n d e x a satisfies an=0, n-i coordinates of z, and the sum S is determined.
z' denotes the first
Let V denote the
(germ of a)
variety d e f i n e d by Zn and the h a , w h e r e an=0. P r o p o s i t i o n 7. Proof.
M is e s s e n t i a l l y
the condition that r(z,~) (8)
finite at 0 if and only if V is trivial.
Note that the condition that r(0,~)
0 = z
+ Zh
vanishes amounts to
vanishes becomes
(z) (~') a
n a If V is not trivial, choose a non-zero w in V. is not essentially finite.
Conversely,
efficient in the p o w e r series vanishes, ishes,
For such {,
and M is e s s e n t i a l l y
finite.
E q u a t i o n 8 then holds
for z=w, so M
if V is trivial, and 8 holds,
then each co-
so z lies in \i.
Since V is trivial,
z van-
101
VI. Ope n questions I.
Prove the implication of IV.4 that in
2.
& (M~p) q
is finite implies that
i
lies
l~(p) .
Give an intrinsic trea~nent of "orders of contact" and "multiplicities" on CR manifolds.
3.
Prove Bloom's conjecture 111.8.
4.
Determine the boundary behavior of the Bergman kernel~ and its relationship to these invariants.
5.
See [He] and [DFH].
What are the important invariants for subellipticity in case pseudoconvex?
6.
M
is not
See [Ho].
Determine approach regions for the boundary behavior of holomorphic functions on weakly pseudoconvex domains.
See [NSW] for the situation when
&n_l(M~p)
is finite. 7.
Find normal forms for the defining equations of real analytic hypersurfaces with degenerate Levi form.
See [D4].
Bibliography [BJT]
M. S. Baouendi~ H. Jacobowitz and F. Treves~ On the analyticity of CR mappings, Annals of Math. 122 (1985) 365-400.
[Be I] S. Bell s Biholomorphic mappings and the ~-problem~ 103-113. [BI]
Annals of Math. 14 (1981)
T. Bloomj On the contact between complex manifolds and real hypersurfaces in E 3 Trans. A.M.S., Vol. 263, No. 2 (1981) 515-529.
[B2]
Remarks on type conditions for real hypersurfaces in ~n pp. 14-24 in Several Complex Variables~ Proc. of Internat. Conf. at Cortona~ Italy~ 1978.
[BG]
T. Bloom and I. Graham~ A geometric characterization of points of type on real hypersurfaces~ J. Diff. Geometry 12 (1977) 171-182.
[ CI]
D. Catlin~ Necessary conditions for subellipticity of the ~-Neumann problem~ Annals of Math. 117 (1983) 147-171.
[C2]
, Boundary invariants of pseudoconvex domains~ Annals of Math. 120 (1984) 529-586.
[C3]
• Subelliptic estimates for the ~-Neumann problem on pseudoconvex domains ~ preprint.
[DI]
J. D'Angelo~ Real hypersurfaces, orders of contact~ and applications~ Annals of Math. 115 (1982) 615-637.
[D2]
, Subelliptic estimates and failure of semi-continuity for orders of contact~ Duke Math. J. 47 (1980) 955-957.
[D3]
, Intersection theory and the ~-Neumann problem, Proc. of Symposia in Pure Mathematics (1984)~ Vol. 41, 51-58.
[D4]
, Defining equations for real analytic hypersurfaces, Vol. 295~ No. i~(1986) 71-84.
[D5]
m
Trans. A.M.S.
, Iterated commutators and derivatives of the Levi form, preprint.
102
[ DF]
K. Diederich and J. Fornaess~ Pseudoconvex domains with real analytic boundaries~ Annals of Math. 107 (1978) 371-384.
[ DFH]
K. Diederich~ J° Fornaess and G. Herbort~ Boundary behavior of the Bergman m e t r i % Proc. of Symposia in Pure Mathematics (1984)~ Vol. 41~ 59-67.
[G]
P. Greiner, On subelliptic estimates of the ~-Neumann problem in J. Diff. Geometry 9 (1974) 239-250.
[He]
G. Herbort, The boundary behavior of the Bergman kernel function and metric for a class of weakly pseudoconvex domains of ~n ~ Math. Z. 184 (1983) 193-202.
[~o]
L. Ho~ Subellipticity of the ~-Neumann problem on non-pseudoconvex thesis~ Princeton University~ 1983.
[H]
L. H~rmander~ The Analysis of Linear Partial Differential Operators III~ IV, Springer-Verlag~ 1985.
[KI]
J. Kohn~ Boundary behavior of ~ on weakly pseudoconvex manifolds of dimension two~ J. Diff. Geometry 6 (1972) 523-542.
[K2]
, Subellipticity of the ~-Neumann problem on pseudoconvex domains: sufficient conditions~ Acta Math., Vol. 142 (1979) 79-122.
[K3]
, Pseudodifferential Symposia in Pure Mathematics
[K4]
Estimates in Pure Mathematics
for
~b
(1985)~
operators and hypoellipticity~ (1973), Vol. 23~ 61-69. on pseudoconvex manifolds~
E2
domains~
Proc. of
Proc. of Symposia
Vol. 43, 207-217.
[NSW]
A. Nagel~ E. Stein and S. Wainger, Boundary behavior of functions holomorphic in domains of finite type~ Proc. Nat. Acad. of Sci. 78 (1981), No. ii~ 6596-6599.
[S]
I. Shafarevich, York~ 1977.
[r]
A. Talhoui~ Conditions suffisantes de sous-ellipticite pour Sci. Paris~ t. 296 (1983)~ 427-429.
[W]
H. Whitney~
Basic Algebraic Geometry~
Complex Analytic Varieties~
Springer-Verlag~
Berlin and New ~ ~ C. R. Acad.
Addison Wesley Publishing Co.~ 1972.
Iterated Commutators
and Derivatives
of the Levi Form
* John P. D'Angelo University of Illinois Urbana, Illinois 61801 Introduction The theory of weakly pseudoconvex conditions
[1,2,3,4,5,6,7].
domains
in
~n
Levi form; all involve taking higher derivatives. notions
come from algebraic geometry,
analytic varieties with the boundary cerned with differential
often involves
There are many ways to measure
study of
theory of complex
In this paper~ however,
geometric notions,
notion first arose in Kohn's
Some of the recently popular
such as the intersection [5].
finite type
the degeneracy of the
we will be con-
such as iterated commutators.
~ [7].
This
Our setting will be CR manifolds;
these are real manifolds whose tangent spaces have a certain amount of complex structure.
We consider two type conditions
point
The first~
p .
its complex conjugate direction", space [1,7].
See also [6].
(L~p) , measures
that are required
the tangential
direction
The second~ written
differentiating
notions
type
M
L
at a
of
L
defined by
that various
L
invariants
a related conjecture
is the following.
Suppose
on the span of
and
in the so called "bad tangent
C(L~p) , and first defined by Bloom [i]~
that the Levi form is semi-definite its iterated commutators C(L, p)
the number of commutators
involves
and its conjugate• formed from these
is a pseudoconvex hypersurface
In this paper we formulate This conjecture
vector field
that is not part of the holomorphic
Bloom has conjectured in case
(I,0)
to obtain a component
the Levi form in the directions
are equal,
field•
for a
in
~n .
for the case of a single vector
that
M
is a CR manifold,
L , and the
up to sufficiently high order.
(I, 0)
Then type
and
parts of all
(L~p)
equals
.
F o r CR m a n i f o l d s pseudoconvexity
be derived
of dimension
hypothesis •
3,
from the work of Bloom [i].
dimension 5.
the result
is
Such hypersurfaces
type, but the general
In this paper we obtain some partial results• conjecture holds for vector fields of type 4. fields of type 2, and in the pseudoconvex
to equal
Secondly, N .
a nd d o e s n o t in
~3 ,
we give an estimate
A simplified
for
case~
require
the result
are CR manifolds
It is also easy to verify this for certain particular
such as a vector field of minimal
type 3.
trivial
For pseudoconvex hypersurfaces
a can
of
vector fields,
case is open.
First of all, we show that the
The result is trivial for vector there cannot be vector fields of
C(L~p)
in case type
(L,p)
is known
form of this estimate gives the result that
Partially supported by the NSF Grants MCS-8108814(A04) by The Institute for Advanced Study.
and NMS-8501008
and
104
C(L~p) ~ max(N~2N-6) commutators~ field and
.
To prove this result, we write out formulas for iterated
and identify the terms as Levi forms applied to the original vector
(I~0)
parts of its higher commutators.
By differentiating appropriate
minor determinants of the Levi form at points of degeneracy~ we are able to obtain an estimate for
C(L~p)
.
See theorem 22.
At no time in these proofs do we make any choice of coordinates. structure is used.
Only the CR
The formulas here could perhaps be of use in problems on weakly
pseudoconvex CR manifolds~ Diederich-Fornaess
because they do not appeal to such theorems as the
theorem [6].
I acknowledge helpful discussions with Tom Bloom on many of these ideas. In fact the motivation for this paper comes from trying to prove some of the conjectures and results in [I] without using his reduction to the homogeneous
case.
I also thank
J. J. Kohn for originally leading me to the study of iterated commutators.
Preliminaries A CR manifold is a real manifold whose tangent spaces have a certain amount of complex structure. let
TM ~ E
To state this more precisely~
there is an integrable subbundle~ I.
let
denote its complexified tangent bundle.
The intersection of
TIOM ~ of
TIOM
M M
JIM ~ 6
be a real manifold~
and
is called a CR manifold if
with the following property:
with its complex conjugate bundle
T0~
consists
of the zero section alone. Henceforth in this paper we will also assume that the bundle sum has codimension one in bundle.
Thus
M
TM ~ 6 .
This bundle sum is called the holomorphic tangent
is an abstraction of a real hypersurface in a complex manifold.
that case local sections of
TIOM
the
are local coordinates.
~/~gj ~
TIOM~ (I~0)
TIOM @ T O ~
where the
z.j
vector fields.
In
are those vector fields that are combinations of
We denote by
~I0
and
We call local sections of ~01
the projections onto the
respective subbundles. To describe our results on iterated commutators~ Levi form.
we recall the definition of the
Notice first that there is a purely imaginary non-vanishing one form
defined up to a multiple~
that annihilates
2. Definition
be local sections of
Let
LjK
~
the holomorphic tangent bundle. TIOM .
We define the Levi form~
as
usual~ by the formula X(L,K) = [~, [L,~]} Here
[,
]
(2.t)
.
denotes contraction and
[ , ]
denotes the commutator.
By the Cartan
formula~ we have
),.(e,'~) = [-d'rl, L^K} Let
L
be a (I~0) vector field.
an iterated commutator of the form
(2.2) We write
L *m
for any vector field that is
105
L *m = [...,[[LI, L2],...Lm] We w r i t e ~
L. is either L or J for any vector field that is of the form ~I0(L ~n)
m
, where each
~ . or its conjugate.
We will also need a notation for orders of vanishing of functions with respect to L
and
at
~ .
p .
Let
p
be a point in
M , and let
f
be the germ of a smooth function
Put
VL, p(f) = vL(f ) = j , if
j
is the smallest integer such that there is a monomial ~ J
of order
j , for which ~ J ( f ) ( p )
any differential definition
of
operator of order
v L , however,
can be of lower order. larger than
X = [L,L]
that the type of
Let
L
L
at
[L*m,~](p)
~ ,
L
and
~ .
for
In the
This is because commutators vL(f )
is VL
If, also,
0 < f < g , then
o
vL(g) ~ vL(f)
.
on (i~0) vector fields.
is
L *m
m , if
m
TIOM
defined near
is the smallest positive
p
We say
integer for which
such that
exists, we say that type
5. Definition.
Let equals
L
(L,p)
equals infinity.
be a local section of
m ,
if
These type conditions gate their relationship.
VL, p(k(L,L))
TIOM
= m-2
are not identical; Obviously
number can be three without that case.
and
is non-zero.
m
C(L,p)
L
We state as a remark two facts about
be a local section of p
there is a vector field
that
.
vL(fg) = vL(f ) + vL(g )
4. Definition.
in
we also write ~ J
that is formed from
we only allow monomials.
We recall two type conditions
If no such
j
Finally~
The reader can exhibit easily examples where
Vx(f) , for
3. Remarks.
is non-zero.
3
defined near
p
We say
.
our purpose in this paper is to investi-
the numbers are simultaneously
the other number being three.
two, but either
Let us briefly consider
We can write
[e,~] = A + ~ + ~(L,~)T , where
T
is purely imaginary,
and [T,~] = I
(6.1)
Then we have [[[e,~],L],~] Thus if
I(L,~)
6.2 enables
= -k(L,~) -L(~(e,~)) vanishes
at
+ ~(L,~)[[T,L],~]
.
(6.2)
p , we see that the first term on the right side of
the whole expression
to vanish at
p
without the second term vanishing~
and conversely
the vanishing of the second term does not in general force the vanishing
of the first.
If, however,
if
~(L,~)
tary facts: k(L~B)
does~ so neither
k
a first derivative
must vanish if
X(L~)
be used repeatedly below.
is assumed to be semi-definite,
type can be three in that case.
then 6.2 must vanish
Here we use two elemen-
of a function at a critical point must vanish, does, when
i
is semi-definite.
and
These facts will
106
7. Definition.
M
is called pseudoconvex
We can always choose the sign of
~
at
p
so that
%
if the form
~
is semi-definite.
is positive semi-definite.
The presence of the third term on the right of 6.2 suggests the following one form~ depending on 8. Definition.
If
X
the definition of
T .
is any local section of
JIM ~ ~ , we put
[~,X] = [[T,X- {X,~}rLn ] . We have omitted [~,X]
as
the dependence on
~X "
This
We will usually write
~ r m was defined by the author
We give an alternate description of this form as
even though we do not need this result. =-~fT~ = minus the Lie deriv, of
9. Proposition. Proof.
from the notation. is real.
Note also that
in [3], but has been little used. a Lie derivative~
T
[~fT~,X]
= [iTd ~ + d iT~,X } = [ d ~ T A X ]
~
in the direction
T
4
+ X(iT~) = { d ~ , T A X }
= T{~,X] - X{~, T] - [~, [T, X]] = T[~,X] - {~,[T,X]]
= -{~,x] We have used the usual formula for Lie derivatives~
the fact that
stant;
again~
of
the Cartan formula~
the fact that
[~;T~ : i
is a con-
. We are now ready to write down formulas
Let
iTS= i
and finally the definition
L *m
denote an iterated commutator
L*m = [...[[LI, L2],...Lm] where each
L. i
is either
for a general iterated commutator.
of the form
,
L
or
~ .
We assume that
LI= L
and
L2=~
.
Then we
have i0. Proposition.
[L*m~]
equals
(~L-L){L*m-I,~) - X(L,~01L*m-I)
(10.1)
or
(10.2)
(~--~)[L*m-l,~} + X(~10L'~n-l,g) We have i0.i if the last vector field Proof.
Suppose for concreteness
L
L
that
L*m-I = ~ 1 0 L e m - i + ~01L*m _ i + [L~m-I raking the commutator of 10.3 with noting the definitions ii. Corollary.
of
Suppose
~
and
that
L
then have the formula that Z((~L-L )(~L-L))j~(Xj,i) In this formula~
is
m m
L
equals
and 10.2 if it is L .
~ .
We can write
]]T .
(10.3)
L , using the integrability
of
TIOM , and
)~ , we obtain the desired formula !0. i. and
~
alternate
[L*2k+2~]
]
equals
in the above commutator.
+ ~((~L-L)(CtL-L))J(~-~))~(L, Yj)
the first sum runs from 0 to
.
k , the second from
We
(ii.i) 0
to
k-l, and
107
X. ]
the vector fields
Xk = L ,
and
Y. ]
are defined as follows.
Xj=~Io[L*2k-2J,L]
for
j
less than
k .
(11.2)
Yj = _~o1L*2k-2j Note the minus
(11.3)
sign in the definition of
Y. . ]
12. Corollary. , that
[L*m
For any iterated con~nutator
] =~m-2-j(gj)
gj = ~ ( L , ~ J + l ) o r Proof.
~
a conjugate
This arises because
L *m
o-Z
~
there are operators k
where the sum extends from
0
~L
to
so
m-2 , and
thereof.
This formula follows by induction from proposition
absorbed the
the (i,0) vector
k.
field arises first in the definition of
into the o p e r a t o r s ~
Note that we have
i0.
m-2-j
Results We can now prove that, under a pseudoconvexity equal
to four simultaneously.
because it motivates 13. Theorem.
Let
on the span of if
We present
hypothesis,
the two types are
this proof in perhaps excessive
M
be a CR manifold.
L,~I0[L,~]
3 and
Suppose
~I0[[L,~],L ] .
that the Levi form is semi-definite Then, type
(L~p) = 4
tion,
Suppose first that
C(L~p)
equals
four.
and the fact that the type is not two~
and vanishes
at
p .
A 2 + B 2 + i[A,B] ~L%(L,~) (p)
.
Letting Since
Because of our positivity assump-
the function
L = A + Bi ~ we see that
[A~B]
%(L,~) ~L
then both
and the formula that
with both derivatives
is first order, we see from calculus
being
L
contraction
A2
and
B2
applied to
L 2 = A2-B2+i(AB+BA) must also vanish.
{[[[L,i],L],~],D}(p)
Thus the type of it is not three,
L
will be four,
that
k(L~)
equals
This becomes,
vanish at
p .
, we see that the analog of 14
Since
C(L~p)
is four,
this
We now show that
the sum of 14 and a non-negative
term.
since it is assumed not two~ and we have seen that
just before definition
we use formula 10.2.
as
(14)
cannot be, so we must have that 14 is actually strictly positive.
7.
To verify the value of the contraction,
after ignoring those terms that obviously vanish
p ,
[L
*4
,'Q](p) = (¢~L-L)(C~,L-L)k(L,~) + (aL-L)X(L ,-~OI[L,i]) = (~LX(L,~)
rToI[L,~]
+ ~.(Trlo[[L,~],L],~)
+ ~k(L,~ol[L,~]))(p)
We must show that the second term in 15 is non-negative. for
is non-negative
can be expressed
is non-negative.
If 14 vanishes, From this~
at
if and only
C(L,p) = 4 .
Proof.
me
detail~
the proof of theorem 22.
and use the Jacobi identity.
This gives
To do so~ we write
(15)
108
{[[L~],i]~}
= -[[[X,L],L],~} -{[[L,L]~X],~] (16)
The first term on the right side of 16 vanishes~ this space.
because
The second term is of course non-positive.
hand side.
is semi-definite
on
We compare with the left
It becomes
-~X(L~X) + terms that vanish at We obtain from 16 and 17
that
(17)
p . must be non-negative
~X(L,~)
at
p . This completes
the first half of the proof. Now we suppose that or else the analogous
type (L,p)
equals four.
We have the right side of 15,
term arising from a commutator of the form
First we assume the the right side of 15 is non-zero possibility
arising from the analogous
15 is non-zero positive.
at
commutator
at
Z = [[[L~]~L],L].
p ; below we show that the
is not actually possible.
So~ if
p ~ we claim that the first term on the right of 15 is itself
Consider
the 2 by 2 determinant 2
f = k(L,~)~(X,X)
- IX(L,~)I
(18)
Because of the non-negativity p ~ so by calculus ~Lf
of
k ~
(and the remark before
is non-negative
at
f
is non-negative.
from the commutator
from 20 we obtain that L2k(L~)
.
L%(L,~)
that this must vanish when
Z
vanishes
~LX(L~)(p)
cannot occur.
vanishes.
~L~(L~)
does~
does not vanish at
arguments
show that,
if
M
p ~ so
at
p ~ that (20)
must be positive. For~ if
~Lk(L~)
Thus the contraction
C(L~p)
is pseudoconvex~
C(L~p)
vanishes~
{Z,~]
then
equals
can be non-zero
type (L~p)
equals four also. then
The remaining
Since we have seen earlier
this contraction
Thus we have proved that if
~Lk(L~)
at
(19)
This is the first term in the analog of 15.
when 15 is non-zero.
vanishes
p .
this all together tells us that
possibility
f
14) we have
This gives us~ if we also assume that ~Lk(L,~) 2 2 l~k(L,~)l + IL~(L,~)l vanishes at p . Putting
Also
is four~
only
then
Note that similar
is always even if it is
finite. We now turn to the general case. 21. Formula from
L
and
gj = X ( L ~ Remark.
[e
*N
,~} = ~ N
~ ~ and j+l)
gj
-2-j (g j)
First we recall the formula in Corollary ,
are functions
or a conjugate
where ~ J
i on
L
or
of order
j
formed
that can be written in the form (21.1)
thereof.
Note that the total number of derivatives
the operator,
are operators
12.
in each term equals
N ;
j
on
and the rest coming from an earlier iterated commutator.
109
We now prove a theorem that gives a bound for type (L~p) that
is finite.
If
type (L~p)
C(L~p)
when we know that
N ~ the result tells us in particular
c(L~p) < 2N-2 .
22. Theorem.
Let
the Levi form
)~ is semi-definite
L be a
its iterated commutators Let S
equals
L *N j ~
(i~0)
vector field on a CR manifold
up to order
to order
j= i
j
to
VL()~(~J+l
~j+l)
L *N .
C(L~p) < max((2N-2-2j-vj),N) C(L~p) < max(N~2N-6)
Otherwise~
Suppose
(I~0)
that
parts of
parts of the iterated commutators
let
v. J
up
denote the number
.
fact~
If
(i~0)
Finally~
Then~
type (L~p)
21 does not vanish.
M .
and all the
Suppose that type (L~p) equals N . *N such that {L ~]] (p) does not vanish. Let
N-2 ~ denote the
coming from
L
N .
be an iterated commutator for
Proof.
on the span of
< 2N-2 .
(The
max
is taken over all
j ). In
.
equals
N ~ then some term in the expression
in proposition
If this is the first term, we obtain immediately
for the appropriate
term,
consider
the inequality
that
C(L~p)
(as in the proof of the
type 4 case) that 0 _< I X ( L ~ J + I ) I
2 < k(L,~)k(~z + I ~)J +~l,~z~"~ J
From this inequality~
since V L X ( L 3 ~
--j+l
(23)
) < N-2-j ~ and using remark 3~ we obtain
the fact that Ve(~(L,~)) < 2(N-j-2)vj
.
(24)
(Note that when
j = 0 ~ we get that
24.)
since
Otherwis%
Since we are guaranteed over
j
to conclude
the case
if
that only one term does not vanish~
the desired formula.
Suppose that
To improve it to
type (L~p)
equals 6.
C(L~p) < max(8-Vl~6-v2~4-v3~2-v4~6)
type (L~p) _> 4 ~
at most 6. C(L, p)
< N ~ by adding
then
to both sides of
we must take the maximum 2N-6 , we first look at
vI > 2 .
Since it must be even~
.
Therefore,
The result of theorem 22 tells
However~
we have already seen that,
we actually have that
C(L~p)
is
and~ by theorem 13~ it is not 4~ we have that
equals 6.
This now enables us to obtain the final statement only term not already vI
v0
type (L~p) = 6 .
25. Example. us that
VL(X(L~))
C(L~p) = (left side of 24) + 2 ~ we obtain the desired formula.
< 2N-6
is the term when
is at least 2~ the result follows.
j= i .
We believe
in the theorem 3 because
the
By including the fact that
that this process can be carried
out further 3 to obtain 26. Con.lecture.
If
M
is pseudoconvex~
the assumption of pseucoconvexity parts of iterated commutators.
then
type (L,p) = C(L~p)
.
More generally,
is only required on the span of all the
(i~0)
110
This conjecture would follow if we could prove the analogous statement for higher orders than four. Suppose that
What we would have to show is the following:
C(L~p) > N ~ and that
M
is pseudoconvex.
Then~ for all possible
choices of iterated commutators and resulting operators ~ j + l VL~Q~+I,y
j+l)
is at least
Thus the Levi forms of these
N-2j-2. This must hold for O_<j_.
(27)
parts of iterated cormnutators must vanish
to various orders themselves. 28. Len~na.
Suppose that
M
is pseudoconvex~
each term in the formula 21.1 vanishes at Proof.
and that 27 holds.
Then we have that
p ~ and hence we have that
type (L~p) > N.
This can be derived analogously to 24~ or we can simply note that if the
result failed to be true~ and if of theorem 22 would give 29. Remark. manifold~
If
v. were J C(L~p) < N .
_> N-2-2j ~ the estimate in the conclusion
M is a real hypersurface of
~n
and
L
is tangent to a complex
then it is easy to see that the only terms in the contraction of an iterated
bracket are those of the form ~ J x ( L ~ )
~ so the result holds.
fields~ some pseudoconvexity hypothesis is required. role here.
For general vector
Pseudoconvexity plays a subtle
See the examples in [I].
Bib liograph Z I.
T. Bloom~ "On the contact between complex manifolds and real hypersurfaces in Trans. A. M. S.~ Vol. 263~ No. 2~ Feb. 1981~ 513-529.
2.
D. Catlin~ "Boundary invariants of pseudoconvex domains"~ Annals of Math. 120 (1984) ~ 529-586.
3.
J. D'Angelo~ "Finite type conditions for real hypersurfaces"~ 14 (1979)~ 59-66.
4.
, "Points of finite type on real hypersurfaces"~
J. Diff. Geom.
(preprint).
5.
~ "Real hypersurfaces, of Math. 115 (1982)~ 615-637.
orders of contact~ and applications"~
Annals
6.
K. Diederich and J. Fornaess~ "Pseudoconvex domains with real analytic boundary" 3 Annals of Math. 107 (1978)~ 371-384.
7.
J. J. Kohn~ "Boundary behavior of ~ on weakly pseudoconvex manifolds of dim 2 "~ J. Diff. Geom. 6 (1972)~ 523-542.
PLURISUBHARMONIC
FUNCTIONS
ON
RING DOMAINS by
i)
John Erik Fornaess
i.
and Nessim Sibony
Introduction.
Griffiths
[GI] showed in 1971 that holomorphic
~n, n _>- 3~ to a compact Kahler m a n i f o l d Shiffman
maps from a punctured ball in
extend m e r o m o r p h i c a l l y
across the puncture.
[SI] extended this result to ~2 (See also Siu [SIU2] and Sibony
[SIBI]).
key step in the proof was to pull back the Kahler form ~ via the holomorphic
A
map f.
t
Then finiteness
of the volume of the graph of f reduced to the estimate
Since there always is a C ~ plurisubharmonic
function u on •
(shrinking
)n <
oo.
function u on the punctured ballJ "7~ such
that ddeu = f ~, the estimate reduced to whether
harmonic
J(f*
the radius
ddCu) n < 0 for any
if necessary).
plurisub-
This estimate was
shown to hold. Griffiths
[GI] proposed to study extendability
K~hler manifolds~
of holomorphic maps,
across balls rather than isolated points.
theorem this follows if the target
is complex projective
into compact
(By the Hartogs
space).
extension
This leads to the
t
question whether ](ddCu) n < ~ where K is a closed ball in ~ n taining K and u is a smooth plurisubharmonic are far from able to do this. monic functions
convex compact
Smoothing of plurisubharmonic
is known that smoothing
We
about plurisubharor a
set.
In section 2 we show integral estimates
W e show that smoothing
of ~-K.
~K = ~-K where K can also be a closed polydisc
show for example that u is in LI(~K),
3.
function on a n e i g h b o r h o o d
But we have made some observations
on ring domains
general polynomially
~ is an open ball con-
funhtions
is always possible
is impossible
that are weaker than the above.
see Theorem 2.2 for a precise on ring domains
We
statement.
is discussed in section
on a general class of ring domains.
in some eases on smooth bounded domains
It
in ~2
( IF1]). A plurisubharmonic harmonic
subextension
function
to ~
such that ~ -< p on ~K' harmonic
is said to have a plurisub-
if there exists a plurisubharmonic function
By Bedford-Burns
extension to B .
and K is a ball.
p on a ring domain ~ = B - K
[BI] we cannot
W e don't know whether
a on ]3,
a ~-~,
expect p to have a plurisub-
subextensions
exist when p is smooth
But in section 4 we give an example of a smooth p which does not
admit a subextension,
K being a po±ydisc
p w h i c h does not admit a subextension~
and in section 5 we describe a discontinuous
K being a ball.
i) Author has been supported in part by an NSF gran~.
112
2.
Integral estimates.
Let ~K = ~ function on ~K"
K be a ring domain in ~ n Our first observation
n > i, and let p be a plurisubharmonic
is that p is bounded above in a neighborhood
of K. Lemma 2.1.
If K is polynomially
convex, there exists a nei~hborhoo d V of K, V c ~ ,
such that sup p(z) < ~. zsV-K Proof.
Fix any open neighborhood V of K, V cc B,
exists a holomorphic
the complex analytic variety Z = {ze ~ n maximum principle
and let peV-K.
polynomial P(z) such that P(p) = i and
for plurisubharmonic
p(p) < sup p(z) < sup zcZ~n~V ze~V
P(z) = i}.
Then there
IPI < i on K.
Let Z be
Then Z.A K # ~ and so, by the
functions,
p(z) < o~
It is Clear from the proof that it suffices to assume that K is n - i meromorphically convex, i.e. that for every point p£ ~ - K there exist holomorphic n-I on 1~ such that Z Ifil > 0 on K and fi(p) = 0, i = l,...,n-l. i=l
functions
fl'''''fn-i
Because of Lemma 2.1 we can without loss of generality restrict our considerations to negative plurisubharmonic Theorem 2.2 ~-K,
functions.
Assume u < 0 is ~ u r i s u b h a r m o n i c
where K is a closed ball.
Slul(ddSu)n-~
i
^ ddCl=l
i
2
< ~
and smooth o n_na neighborhood
of
Then
and i f
,
9
= ~(O,r), [(l~l-r) (ddC~) n
<
DK Observe that we can add a multiple of Izl 2 to u.
Applying the first integral
estimate to the new function it follows that u is in LI(~K ) . Proof.
Let p = max {IzI2-r 2, o} and let {pk ~ be a sequence of non-negative
smooth plurisubharmonic
functions converging uniformly to p, and even in C 2 - norm
on compact subsets of ~K' Pk E 0 in some neighborhood Integrating by parts twice we get that
S pk(ddcu)n = S pkdeu ~ (ddcu)n-i ~K
~K
of ~
(0,r), Ok ~ p near ~ .
+
u ddcp,
A
(ddcujn-l,
R~
Hence
i s bounded by a c o n s t a n t independent o f k .
Since both integrands a r e nonnegative, t h e
theorem f o l l o w s . Let u be plurisubharmonic on a r i n g domain RK, K polynomially convex, u f Then ddCu i s a c l o s e d , p o s i t i v e (1,l)c u r r e n t .
--.
Instead of asking f o r a plurisubhar-
monic s u b e x t e n s i o n v o f u we can a s k f o r a c l o s e d , p o s i t i v e (1,l)c u r r e n t S on B which i s a s u p e r e x t e n s i o n o f ddcu, i . e . S
2
ddCu on Rk.
e x t e n s i o n whenever u h a s a s u b e x t e n s i o n .
We d o n ' t know i f ddCu h a s a super-
However, t h e o p p o s i t e i s t r u e .
As i n d i c a t e d
i n t h e i n t r o d u c t i o n we would l i k e t o know i f t h e s t r o n g e r e s t i m a t e
P r o p o s i t i o n 2.3. S 2 ddcu
on RK,
I f t h e r e e x i s t s a p o s i t i v e , c l o s e d (1,l)c u r r e n t S
u ~ l u r i s u b h a r m o n i c on 5 neighborhood
plurisubharmonic s u b e x t e n s i o n v Proof.
of
u
B, v
S u
of
on RK'
There e x i s t s a plurisubharmonic f u n c t i o n p on B such t h a t S
K'
Hence v :
= p-c
with
\ K, then there e x l s t s a
Then ddc(p-u) 2 0 and hence p-u i s plurisubharmonic on O above on OK.
on 1B
=
ddCp([~1~).
By Lemma 2 . 1 p-u i s bounded
i s a subextension o f u i f c i s a l a r g e enough p o s i t i v e
constant.
3.
Smoothing.
We w i l l prove a smoothing theorem f o r plurisubharmonic f u n c t i o n s on r i n g domains R where K i s a s t a r s h a p e d compact s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n : K
114
Theorem
3.1.
Let P be mlurisubharmonic
. . . compact and satisfies .condition plurisubharmonic Proof.
functions
(i).
ring domain 9K : ~ - E
Then there exists a sequence
plurisubharmonic
([RI])
functions
converging
{Km}m= I be a sequence
and plurisubharmonic
of compact
for each m an integer n(m) sup ~m,n(m) mak
is continuous
Since p
~
such that am,n(m)
converge
converging
To see this, assume Let
Em+ I V m and 2 k : UK m.
i ~ Ok,m + m--
and plurisuhharmonic
= max(p,-k)
functions
on ~K and ~ m , n ~ ~m pointwise.
sets in QK' Km c i n t
{0n}~= I
down to o.
down to ~ and we can smooth each ~m' then we can smooth ~. are continuous
~ Cof { Pn )n=l
it suffices to find a sequence
Notice that if {o m }m:l is a sequence of plurisubharmonic
(Om,n)n=l
where K is
such that P n ~ P pointwise when n ÷ ~.
By the theorem of Richberg
of continuous
o~a
on K m k S m .
Choose
Then T k :
on ~KIK and T k ~ ~.
down to p, it suffices b y the above observation
to smooth p in the case p is bounded below. Let (~m) be a sequence of balls, am be a continuous Let p(k):
plurisubharmonic
= sup(p,
and p (k) is continuous
sup ~m ). m~k
K c ~ i c c...c c a m c
function,
on ~ - B-k + I.
d=p on ~ k + l - K
on ~n-E, because
Pn :
{
=
if ( ~ ) n
near
function
max{a,0)
sequence
~.
we can assume
We m a y also assume that
~ on ~n-K such that p is continuous
and o < p on ~ k + 2 - ~ k + l " is a smoothing
and let
on ~k' p ( k ~ p
Using again the same observation
that p is bounded below and is also continuous
~K'
B
~m ~ p on Bm-K and Sm ~ p on B - ~m+l.
Then each p(k) is plurisubharmonic
there exists a k and a plurisuhharmonic on ~ -
c ~m+iCc UBm=
It then suffices to smooth
for ~, then
-
on ~ k + ~ K
on ~ l~k+~
is a smoothing
sequence for p on ~k if we start with large enough n.
We assume then that p is plurisubharmonic a sequence of continuous Fix a star-shaped
C~
plurisubharmonic
function X: ~n ÷ ~
+
En-K,
functions
converging
u (0~ with supp X c ~ (0,i).
condition on K, tKc int K.
and ~ (p,a) c En-tK V P S
on ~n-K and that p ~ -~.
We will find
down to p on ~n-K. Let ts(0,1).
Hence there exists an s=s(t)>0
By the
such that a
We can assume that s depends continuously
on t and that
~(t)~0. For ts(0,1) pt(z) = p(z/t).
ot (z) :
~
let Pt be the plurisubharmonic Also,
define a smoothing
I x(-~)pt(z+~) ~(o,~)
function on ~n-tK defined by pt(z) =
d t of Pt on ~n-K by
115
Then ~t is C , plurisubharmonic
at(z)
and ~t
Pt"
Hence we obtain the crucial
estimate
(2)
~ p(z/t).
If k ~ 2, let pk(z)
= sup ot(z). i 1 - ~ -
We show that
each Pk is continuous
and plurisubharmonic
on ~n\K and that pk~O
pointwise. At first we show
pk(z)
(3)
~ p(z) v k , v zs~n-K.
This follows-from
p(z) = lim
(2) because
p(z/t),
([HEll).
Next we show that
a(t,z):
= - at(z)
To prove
(~,~):
is continuous
(4) it suffices
= ~ x (~--F'U~)
on (0,i) x (~n-K).
to demonstrate
p
continuity
(4)
of
(~)
~$(z,~(t)) but this is immediate
lim ~(t,z)
since ps L I £oc"
= p(z)
(5)
(t,~) ÷ (1,z)
This follows
since
(3) gives
~ and upper
and (5) and the obvious monotonicity
semicontinuity
it follows
that pk~p.
of p gives s.
From (3)
It only remains
to show .
that
each Pk is continuous.
upper regularization
of Pk"
(4) and (5) it follows of each a t it follows (3) and (5) it follows z such that ~t(~)
Plurisubharmonicity Fix k, z and s > 0.
that pk(z) that
0k(U)>
that there
is then automatic
since Pk = Ok' the
We show continuity
of Pk at z.
< ~.
From the definition
pk(z)
- s for all ~ in some neighborhood
exists
< pk(z) + s/2 V w s U
of Pk and the smoothness of z.
a to, i-i/k
and t o _ t < i.
From
Shrinking
U if necessary
From U of we
118
obtain from (4) the same inequality
Pk(m) < p k ( Z )
for ~sU~ l-i/k < t ~ t . o
+ s g wsU a n d c o n t i n u i t y
follows.
We don't ][now if we can smooth on domains bounded s t r o n g l y
4.
~i\~2 if ~2 cc ~i are two smoothly
pseudoconvex domains.
Subextension
across a pol~disc.
We will prove the following theorem. T h e o r e m 4.1.
subextension
Choose first a function gaA~([Izl_>-l])
power series of g diverges We m a y assume that
Let p(z,w)
Next,
choose a nonempty,
= log(If(z)l 2 + lw-g(z)l 2) if Izl > i. lw[ < 1/2 and X(W)-= 0 if lw] > 2/3.
Assume that u has a plurisubharmonic
theorem
v(z,w)
of o at every point
subextension (z,g(z)),
in E = ((z,g(z))~
Remark 4.2. harmonic
~ across ~2(0,i).
Then the
zsE, is at least one.
By Siu's
> i] is a complex-analytic
z~E),
every point
If u
n
to approximate
on ]~ (0,2)=~2(0,i-6)~
+ u on compact
set.
Except
in E is a regular point function.
This implies
finitely many.
to our choice of g.
It is impossible
functions
enough, then u: =
formal power series at every point in E except
Hence we have a contradiction
set E
Choose a C~ fumetion X(W)
If K is large
of X and X is locally a graph over the z-axis of a holomorphic that g has a convergent
compact
on ~ 2 ( 0 , 2 ) - [ 2 ( 0 , I ) .
([SIU i]) the set X ~ {(z,w)~ v(z,w)
for finitely m a n y points
perfect
for an A ~ function f on { I z l ~ l }~ ([C1]).
X(w) p(z,w) + K[(z,w) I2 is plurisubharmonic
Lelong-number
to ]32 (0,2). with the property that the formal
at every bou~idary point of the unit disc.
Igl < 1/3.
in [ I z l = l } w h i c h is a zero-set
with X(w)- 1 if
-2 function u on ~ 2 (0,2) - A (0,i)
There exists a C ~ plurisubharmonic
which does not have a ~lurisubharmonic Proof.
Hence
u on compact
subsets by plurisub-
6 > 0 fixed.
sets, then v: = (l~m u )* Is a plurisubharmonic n
subextension
of u to ~ (0~2)-Z2(0, 1-~). For this remark it w o u l d by Bedford-Burns
5.
suffice to use the nonextendable
functions
constructed
([El]).
Subextensmon
Theorem 5.1.
across a ball.
There exists a ~lurisubharmonic
p ~ -~ w h i c h does not admit a plurisubharmonic
function
subextension
We don't know whether p c~n be chosen to be smooth. p can be chosen such that the trivial
p on ~ 3 ( 0 , 2 ) _ ~ 3 (0,i), to ~ 3 (0,2).
We also d o n ' t know whether
(=0) extension across ~ 3 (0,i) has infinite
Lelong number at one b o u n d a r y point of ~ 3 (0,i). The proof will show that the restriction have a subextension
of p to ~ 3 (0,b) - ~ 3 (0,a) fails to
to ~ 3 (0,b) for any a~ b, i< a < b <
2.
117
Let ~l,...,~n at first
be distinct
complex numbers,
and let 0 < c < < 6 < < i.
show:
Lemma
5.2.
a_n_nirreducible
The set Xs, ~ in ~3(z,~,q),
X
a,~
n = {~ (~-~oz) i i=l
We will
n = s A H(D-~iz)=~}__ i=l
is
c0m~o!ex manifold•
n 5.2. Let X = (H (w-cy..z) = s}, s > 0. Then X is a complex n a i=l i s in ~2 since V(H (~-~iz))~ 0 everywhere except at the origin. i:l
Proof of Lemma manifold
We show that X
is connected. First, observe that X is an n-sheeted branched 6 over ~(z) with branch points a I ..... a k s ~(z). All branch points are non6 Let D s = sup (laj]}. Then ~ ÷ 0 as s ÷ 0. When ]zl > qs , X consists of n j . s
covering zero •
disjoint n > 2.
graphs, If X
{XU_}," j = 1 ..... n, X ~d : {~ = fJ(z)}, E 6 s
has two components,
(z, 61(z)),...,(z, m~itiplicities,
6P(z))
~
fO(z) ~ a
- adz ~ + 0 when
z ÷ ~
if
and X2~ let
and (z, yl(z)) ...... (z, yq(z)
the two components.
Then f(z):
locally
parametrize,
= H (6l(z)-yJ(z))
with
is a
i=l,... ,p j=l,... ,q polynomial
of degree pq without
zeroes.
This is of course
impossible,
so X
is
connected. Next we fix 0 < s < < 6 < < i so that , q
< i.
Let R.(S.)j J be a closed line
the unit circle,
j = l,...,k.
(aS.) N {a.6} = ~. J J
segment
starting
We may assume that
at a~ (aj) and ending on
We can choose the line segments
and so that i is on none of them.
Let
to be pairwise
{(i, 6i,~.)} n be the points J i,j=l
disjoint
in Xe, ~ with
z = 1. We consider k
are {a~.}
Xs, @ as a branched
cover over the z-axis.
in r~(z)
6 k
u {aj}
J j:l
j=l
We show that Xc, ~ is a complex manifold. several
The branch p6ints
cases.
branch point
If (zo, Wo) is not a branch point
for X5 then we can parametrize
holomorphic.
If (Zo, Wo)
can be parametrized
is a branch point
as {(z, g(z))]
nearby.
{(f(w),
w)}, hence Xs, 6 can be described
applies
when
(Zo, D o) is a branch point
To show that Xs,6 is irreducible, any two points
If (Zo, Wo'
in {(i, ~i' ~j)}"
Xs, 6
qo ) s Xs, ~ we consider
for Xs and (Zo' Do) is not a as {(z, f(z),
for Xc, then z In that case X
by ((f(~),
o
g(z))}
nearby,
f, g
= as. for some j and X 6 j can be parametrized
~, g(f(~)))}.
A similar
as
argument
for X 6. it suffices
Fix a ~j and ~il,
to find curves Cm2. .
in X
By summetry
5 connecting it will suffice
118
to find a curve connecting
(i, gil, ~j) to (i, ~i2, ~j).
Let y be a curve in Xs from ~il to gi2. y to the z-plane,
We can assume that the projection 6 (aS}, {a.}. J J
X, avoids all branch points
A to XE,6 which starts at (i, ~Zl., ~j).
Also,
Then I has a unique lifting
A ends at (i, ~i2, ~ )
If A ~ j, we will change y so that A ends at (i, ~i ' ~j)" 2 that I is contained
R.'s or S.'s on the unit circle. J J
We m a y at first assume
lines.
We can assume that I only intersects
For every corner of i in the open unit disc add
and subtract a smooth curve going to the unit circle without the straight line segments
following property: m a n y curves
for some ~.
in the closed unit disc and consists of finitely many arcs on the
unit circle and finitely m a n y straight
Subdividing
intersecting
any R. or S.. J J
further at first we can assume that i has the
I consists of finitely m a n y arcs on the tmit circle and finitely
{I.i } starting and ending on the unit circle.
Furthermore
each }~. l can be
contracted to the unit circle T or to T u S. or to T u R. for some j without J J secting any (other) If X.I contracts
S k or R k.
If X. contracts l
to T u S j, contract
A so that A ends up at (i, ~i2, ~j).
to R, replace
li further to T.
inter-
I. by such a contraction. i
These last contractions
change
Hence we have shown that Xc, 6 is irreducible.
Observe that if s = 6, n >- 2, then XE, 6 is reducible. n by the equations { H ( ~ - ~ i z ) = ~ and q = ~ ] . i=l m G1 ~ •
of
One component
is given
Proof of Theorem ~.i. For each integer m > > i choose distinct complex numbers m ,~Nfm~~ such that for all S m ' 6m .> 0 small enough ' all points in ~3(0,2)-]B3(0,I)" " -- " " " "
are closer to Xm:
= {H (~-aiz) = s i=l
^ N( i=l
-~ z)=6
]
than exp.-m3,.(]
m
that ~i = 0 for all m and that sm < < 6 < < 1 as in Lemma 5.2. m gm, 6 are small enough for the following m as a graph,
to hold.
m
Define a plurisubharmonic
h
m
~nd @m(Z,
w,q)
= T m
on ~ ( 0 , 2 ) - ~ ( 0 , i )
1 ,n([~[2
by
We also assume that
We can describe
I~I < 3.
q : gm(Z), ~ = fm(Z), valid for ½ <
--~--A~(l~-fm(=)l 2 + ]~ -gmll}
We m a y assume
m
X
m
near
Set ~m(Z, ~,~) +
{q= w =0}
=
1 <1~]2+1~1 2 Inl 2) + ~-m
)-~--. m
119
f @m if I~ol2 + i,~ 12 <
1 2 2:m
h
max {$m' Sm } i f ] -
2
m
< ~
1~[2 + ]sl2
< 2 ~ 2
2m
~m
Let p: = Z h . m
if [~12 + I~12 >
m
2 m
Then p is p l u r i s u b h a r m o n i c on B ( 0 , 2 ) - ~
(0,i), p ~ -~.
We will show
that p has no subextension. A s s u m e that ~ is p l u r i s u b h a r m o n i c on ~ 3 (0,2), s ~ - ~ and a ~ p Fix an m.
Let Y
m
consist of the points in ~ ( 0 , 2 )
By Siu's t h e o r e m (ESIUI]), Y
m
on ~ 3 ( 0 , 2 ) _ ~ 3 ( 0 , i ) .
where the L e l o n g - n u m b e r of p ~ i/m 2.
is a c o m p l e x analytic v a r i e t y a n d b y construction, Y
contains a n o n e m p t y r e l a t i v e l y open subset of X . m
Hence Y
m
m
contains X . m
If B is p l u r i s u b h a r m o n i c on B ( 0 , r ) , p ~ K and the L e l o n g - n u m b e r of p at 0 is at least c
> 0, then ~ ~ c loglz I + K = clogr.
Fix a point p in ~ ( 0 , 2 ) - ~ ( 0 , i ) .
Then
t h e r e exist K, r > 0 such that for all l a r g e enough m,
~(p) ~ I
log(exp(-m3))
+ K -
m
! 2 logr m
H e n c e o E - ~ c o n t r a r y to assumptions. R e m a r k 5.3.
As follows from p r o p o s i t i o n 2.3 the current T = ddCp does not have
a s u p e r e x t e n s i o n as a closed current in B 3 (0,2).
But if S is the current of integ-
ration on an analytic h y p e r a u r f a c e in ~ 3 ( 0 , 2 ) _ ~ 3 (0,i), then S extends to a closed current in ~ 3 (0,2) (because the v a r i e t y extends). It is easy to construct a similar example in 3 2 ( 0 ~ 2 ) - ~ 2 (0,i). the analytic
sets (of dimension one) do not extend in general.
In that case
120 References
[Bl]
Bedford, E., Burns, D.: Domains of existence for plurisubharmonic functions, Math. Ann. 238 (1978), 67-69.
[C1]
Carleson, L.: Sets of uniqueness for functions regular in the unit circle.' Acta Math. 87 (1952), 325-345.
[Fi]
Forn~ss, J.E.:
Plurisubharmonic functions on smooth domains, Math. Scand. 53
(1983), 33-38. EGZ]
Griffiths~ P.: Two theorems on extension of holomorphic mappings, Inv. Math 14 (1971)~ 27-62.
[HZ]
Harvey, R., Shiffman B.: 99 (1974), 553-587.
[HEll
Helms, L.:
[R1]
Richberg, R.:
A characterization of holomorphic chains, Ann. Math.
Introduction to potential theory, Wiley-interscience
(1969).
Stetige streng pseudokonvexe F~iktionen, Math. Ann. 175 (1968),
251-268. [SZ]
Shiffman, B.: Extension of line bundles and meromorphic maps, Inv. Math. 15 (1972), 332-347.
[SIB1]
Sibony, N.: Quelques problemes de prolongement de courants en analyse complexe, Duke Math. J. 52 (1985), 157-197.
[SIU1]
Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Inv. Math. 27 (1974), 53-156.
[SZU2]
Siu, Y.T.: Extension of meromorphic maps to K~hler manifolds, Ann. Math. 102 (197~), 421-462.
Characterizations of Certain P s e u d o c o n v e x Domains w i t h Non-Compact Automorphism Groups
Weakly
by R o b e r t E. G r e e n e * and S t e v e n G. K r a n t z *
*Work s u p p o r t e d
in part
by t h e N a t i o n a l
Science Foundation
122
INTRODUCTION Let
~ C~ Cn
be a smoothly bounded domain. Let
Aut
Q
be the group of
biholomorphie self-maps (or automorphisms) of 9. I t i s a s t r i k i n g r e s u l t of Bun Wong [30] t h a t i f ~ i s s t r o n g l y pseudoconvex and Aut ~ i s non-compact then
9
that i f
i s biholomorphlc to the b a l l .
More r e c e n t l y Rosay [26] has shown
~
QO e ~9
i s a domain, i f there i s a
s t r o n g l y pseudoconvex, and i f there i s a ~J(Po ) -~ QO'
then
~
near which 8~
POE ~
and
~j e Aut 9
is
C2
and
such t h a t
i s biholomorphic to the b a l l . (Observe t h a t in
C1 the
r e s u l t follows from standard unlformization techniques; new methods are needed when n > 1.) I t i s the purpose of the present paper to explore what happens i f , the l a s t sentence, Qo i s a weakly pseudoeonvex boundary point. The question cannot be answered in general a t t h i s time.
But in case
in
QO i s a
s p e c i a l s o r t of "weak type" point in the sense of Kohn [18] and Bloom-Graham [3], then complete r e s u l t s can be obtained (see Theorems 1.1 and 1.2 below). A f u l l understanding of the general type of boundary point QO w i l l involve much more s u b t l e techniques. 14.
A d i s c u s s i o n of what l i e s ahead i s in Section
Section 1 formulates the p r i n c i p a l r e s u l t of the paper: Theorems 1.1 and 1.2. Section 2 introduces the concepts and n o t a t i o n which are needed in the proofs. Section 3 records c e r t a i n f a c t s about a n a l y t i c e l l i p s o i d s . Section 4 c o l l e c t s the three maln t e c h n i c a l lemmas (Lemmas 4.1, 4.2, 4.3) which are needed in the proofs of Theorems 1.1. and 1.2. Lemma4.1 i s proved in Section 6, Lemma 4.2 i s proved in Section 7, and the more d i f f i c u l t proof of Lemma 4.3 i s contained in Sections 8-13. Section 5 g i v e s the proofs of Theorems 1.1 and 1.2, assuming the t r u t h of Lemmas 4.1, 4.2 and 4.3. The t e c h n i c a l lemmas 4.1-4.3 should have independent i n t e r e s t for f u r t h e r i n v e s t i g a t i o n s into mapping problems. Lemma4.1 g i v e s a method for constructing a biholomorphic mapping as a limit of a sequence of "almost" blholomorphlc mappings. Lemma 4 . 2 g i v e s a t e c h n i q u e for showing that two domains are blholomorphically inequivalent. Lemma 4 . 3 r e v e a l s how c e r t a i n btholomorphlc lnvariants depend on boundary data. Also Sections 8 through 12 each
contain
boundary
useful
results
about
dependence
of
biholomorphic
invariants
on
data.
The contents
of
Sections
5-14
are
as
follows:
Section 5. Section 6.
The Proofs of the Main theorems The Proof of Lemma 4.1
Section 7.
The Proof of Lemma 4.2
Section S. Section 9.
Uniform Estimates for the ~ Analysis of Peaking Functions
Equation
Section 10. Uniform Approximation of Holomorphlc Functions Section 11. L o c a l i z a t i o n of the Caratheodary and Eisenman Volume Forms Section 12. Estimates on the Biholomorphie I n v a r i a n t Section 13. The Proof of Lemma 4.3 Section 14. Further Remarks and Speculations
O~
123
It is a pleasure to thank S. Bell, J.E. Fornaess, S. Plncuk, and H.H. Wu for helpful discussions regarding the general subject matter of this paper. The second author thanks Princeton University and the Institute for Advanced Study for their hospitality during a portion of this work.
§1.
Statements
of Results
This section explanation
presents
of their
Theorem i.i:
Let
the statements
particular
form i s
0 < m E Z.
Let
Em = { ( Z l , Z 2 )
Let
D CC C 2
(l) (ii)
be a domain with
of the principal contained
Em c C 2
: [Zl[2
C3
An
3.
be given by
+ [ z g [ 2 m < 1}.
boundary such that
~ = (1,0) ~ an There are neighborhoods
U
such that,
blholomorphism,
up t o a t o c a l
results.
in Sections
of
~
in
~
and U N 3~
V
of
and
~.l
in
5Om
V N ~E m
coincide. If
there
are a point
c j ( P o ) ~ ~_ a s
j ~ ~
Remark:
that
special be
Notice
Poe
fl
then
when
case of Rosay's
and a u t o m o r p h l s m s ~
m = 1
theorem
biholomorphic
near
~
For e x a m p l e ,
biholomorphic (li')
of
to
Em.
i n Theorem 1 . 1 t h e n t h e r e s u l t (of course
and
bE m
becomes a
Rosay does not require
that
our methods). than is stated.
~
In fact,
( i i ) c a n be w e a k e n e d c o n s i d e r a b l y . What we 5fl n e a r ~ is that it is qualitatively like it
suffices
change of coordinates b~
such that
~j
C3, but that extra assumption is necessary for The t h e o r e m i s t r u e i n much g r e a t e r g e n e r a l i t y
as the proof shows, hypothesis actually need to assume about ~E m
is
osculate
to assume that,
near ~
, ~fl
to order
after
a local
satisfies
2m + 1
at
points
of t h e form
(el0,0); and (ii')
The e i g e n v a l u e o f t h e form
In practice variable,
(ii') once
can often
(1t")
o f t h e L e v i form h a s s i z e
~[z212m-1
at
points
(eie,z2). be a r r a n g e d
has been checked.
wlth a conformed mapping In the
zI
124
The analogue techniques formulated
Theorem has
for Theorem
1.I when
n > 2
can be proved by the
of the present apper only in a special as a separate theorem:
1.2:
The result
Cn + l
boundary
of and
Theorem
I.I
holds
Em
the
form
has
for
ehse.
The result
~ c ~n,
n > 2,
is now
provided
£ m = (z E C n : IZll2+ .... +IZn_ll2+IZn 12m < I}.
§2.
Notation
and Definitions
Throughout
this paper,
the symbol
~
will denote a domain
in
~n _ that
is, a bounded, connected open set. The domain Q wlll be said to have Ck boundary if there is a defining function p for ~ which has the following special
properties: (i)
p
(ii)
p
(iii)
Sometimes
it this
from
B
is
on
useful will
into
~,
and
is
Jacobian matrix
< 0},
to
define
(over
~
p
only
in
the
unit
B(~)
the
holomorphic
C)
of
at
or superscripts) or Euclidean
ball,
mapping
F
a neighborhood
of
some boundary
comment.
denote
a holomorphic
subscripts
complex number
: p(z) ~.
be done without
B = Bn ~ C n
F : ~ 1 --~ ~ 2
(without
Ck ,
Vp ~ 0
point
Let
is
~,
~ = {z e C n
(iv)
and
: cn--~
and
z.
~(B)
the
mappings z e ~1'
holomorphic from
let
In what follows,
denotes Euclidean
length of the vector
~
F'(z)
to
mappings B.
denote
the symbol
absolute ~ E C n,
If the
{~[
value of the depending on
125
the context, Now if element of
0
i s a domain and
~ Cn
0
at
is defined
z
~(z)
z ~
t o be
= s u p { I d e t F ' ( z ) I : F ~ B(O), F ( z ) = O}.
L i k e w i s e t h e E i s e n m a n volume e l e m e n t o f E i s e n m a n - K o b a y a s h i volume e l e m e n t )
~(z)
It
= inf[1/[det
O
Also if
~
~1C (z) _~ ~ 2 ( ~ ( z ) ) )
: 01 - ~ 0 2 for all
at
is defined
F'(O)[
z
(sometimes called
by
that
~(z)
_~ ~ ( z ) ~E 1
is holomorphic then z ~ 01 .
the
: F E O ( B ) , F(O) = z}.
i s i m m e d i a t e from t h e d e f i n i t i o n s
z E O.
t h e n t h e C a r a t h e o d a r y volume
It follows that
for all
(z) _~ ~
2 (~(z))
and
the quotient
~(z) %(z) : ~ ( z ) is a biholomorphic invariant blholomorphic then
in the sense that
Q~l(Z) = 0 ~ 2 ( ~ ( z ) )
for all
if
~ : O 1 - ~ 02 z ~ 01 .
is
See [22]
for
f u r t h e r d e t a i l s on t h e s e m a t t e r s . For c o m p l e t e n e s s , and s i n c e t h e y w i l l be u s e d o c c a s l o n a l l y l a t e r , definitions o f C a r a t h e o d a r y and g o b a y a s h i m e t r i c s a r e r e v i e w e d h e r e . Let O be a d o m a i n , z E 0 , ~ E Cn. The i n f i n i t e s i m a l C a r a t h e o d a r y m e t r i c f o r fl a t z in the direction ~ is
F~(z,~) Here
~ sup{[F'(z)(~)[
where
p
general,
at
z
sup
is the Polncare-Bergman metric t h e same a s t h a t
e I ~ (I,O ..... O
~
O)
gotten
~
is
z,w E O
is defined
t o be
p(f(z),f(w)),
on
B.
by i n t e g r a t i n g
then the infinitesimal
in the direction
form o f t h e
: F E B(O), F ( z ) = 0}.
II represents Euclidean length. The C a r a t h e o d a r ¥ d i s t a n c e b e t w e e n p o i n t s
dist~ar(Z,W)
the
This distance F~
is not,
along curves.
in
If
form o f t h e K o b a y a s h i m e t r i c
for
126
F • = (i n f { ~ z: a ,> 0 •and ) 3 F(O) = z
and
F e 9(B)
F'(O)(el)
such that
= ~/a}.
Arc length and distance are constructed from F~ [22]).
The Kobayashi distance in
9
from z
in the usual way (see
to
w is denoted
Holomorphlc mappings are distance decreasing in both the
dlst~ob(Z,W).
Caratheodary and Kobayashi metrics. Biholomorphic mappings are isometries. The following lemma, f i r s t discoverd by Bun Wong but proved in i t s present generality by Rosay, i s the philosphical s t a r t i n g point for what i s done in t h i s paper. Again see [22] for further d e t a i l s . Lemma 2.1:
Let
~ C'C Cn
There i s a
z E9
be a domain (not necessarily with smooth boundary).
such that
Q~(z) = I
i f and only i f
~
i s biholomorphic
to the b a l l . The o t h e r
initial
fact
about
~
and
~
which w i l l
be n e e d e d h e r e
follows from the work of Graham [8]. Graham studies the asymptotic behavior of the infinitesimal form of the Caratheodary and Kobayashi metrics at points z E ~ which approach strongly pseudoconvex boundary points. I t was noted in [13] that the same calculations yield estimates for the asymptotic behavior of
M~, M~ near a strongly pseudoconvex boundary point.
These ideas w i l l
now be reviewed beginning with the notion of s t r i c t l y pseudoconvex.
Definition
2.2:
If
~ ~
Cn
i s a domain and
point of
(Levi) pseudoconvexity
near
satisfying
P
n
complex t a n g e n t
tangent
vector
there
B2p
j,k=l for all
if
vectors
P
is a
C2
then
defining
P
is called
function
(P)wj~k ~ 0
a
for
(2.2.1)
azja~ k w = (wI . . . . .
wn)
at
P,
Here a complex
i s one s a t i s f y i n g
j =~.l ~ 3zj
The p o i n t
P E ~
is said
t o be s t r o n g l y
~.
(P)wj = 0 .
{or s t r i c t l y )
B2P
J , k = l 3 z j a z k (P)WjWk
> 0
pseudoconvex if
(2.2.2)
127
f o r a l l non-zero complex t a n g e n t v e c t o r s
w.
The d e f i n i t i o n s in 2.2 a r e e a s i l y seen to be independent of the choice of defining function.
Indeed the q u a d r a t i c form ( 2 . 2 . 1 ) , c a l l e d the Levi form,
i s the same f o r a l l l o c a l d e f i n i n g f u n c t i o n s
p
which s a t i s f y
[Vp(P)[ = 1.
When r e f e r e n c e i s made below to the " e i g e n v a l u e s of the Levi form a t
P",
these w i l l always be computed with r e s p e c t to a d e f i n i n g f u n c t i o n p
which
satisfies
[vp(P)I
= I.
If
P • ~
there exists a defining function
~
~2~
n
i s a s t r o n g l y pseudoconvex p o i n t then for
_
j,k=l ~)zjbzk for a l l
w e Cn.
If
E ~ ~
p o i n t s then t h e r e a r e a
I f a l l p o i n t s of
(P)wjwk _
and a c o n s t a n t
P • E 5~
near
P
such t h a t
_> cfwJ 2
(2.3)
i s a compact s u b s e t of s t r o n g l y pseudoconvex
~
simultaneously for a l l
~
and a l l
C
so t h a t (2.3) i s t r u e
w E Cn.
See [22] f o r d e t a i l s .
a r e ( s t r o n g l y ) Levi pseudoconvex then
termed ( s t r o n g l y ) pseudoconvex.
Recall t h a t
C2
~
is
Hartogs pseudoconvex domains
a r e Levi pseudoconvex and c o n v e r s e l y . In a l l t h a t f o l l o w s , when a n a l y s i s i s being done n e a r a s t r o n g l y pseudoconvex boundary p o i n t , i t w i l l always be supposed t h a t a d e f i n i n g
f u n c t i o n h a s been s e l e c t e d w h i c h s a t i s f i e s (2.3). Now t h e c o n s e q u e n c e o f t h e c a l c u l a t i o n o f Graham [8] w h i c h i s o f g r e a t e s t i n t e r e s t h e r e may be f o r m u l a t e d a s f o l l o w s . (See [ 1 3 ] , C o r o l l a r y Theorem 3 . ) .
P r o p o s i t i o n 2.4: is a
6 > 0
Let
~ C C ~n
such that if
be s t r o n g l y pseudoconvex,
z E ~
and
dist(z,b~)
< d
then
Let
z > O.
to
There
i S Q~(z) < l+s.
One of the p r i n c i p a l t h r u s t s of the c a l c u l a t i o n s in the p r e s e n t paper i s to see how
6
in P r o p o s i t i o n 2.4 may be e s t i m a t e d in terms of
c e r t a i n boundary d a t a of
~
and
~.
A f i n a l n o t i o n t h a t w i l l be needed i s a n o n - i s o t r o p l c " d i s t a n c e " modeled on the geometry of the Levi form near a s t r o n g l y pseudoconvex boundary p o i n t . Let
~ CX: Cn
have
outward normal to
C2 b~
boundary. at
P.
Let
P e ~
and denote by
Define D p = CUp.
and
~p =(np) t o be t h e
(Hermitian)
orthogonal
complement t o
~p.
Up
the u n i t
128
z E ~
If
then w r i t e
z N E Dp,
z = P + zN + zT
Such a decomposition, o v e r
E,
i s unique.
understood to be d e f i n e d by the e q u a t i o n .
dp(z)
This f u n c t i o n
dp
and
P
6
~p,
The v e c t o r s
zT
and
zN
both l l e in
it
~
~ I ZNI + ] ZTI 2
i s d e f i n e d only f o r it
are
Set
can be considered to measure d i s t a n c e , but i t
sense a m e t r i c because i ) z
zT
P E ~
i s not symmetric.
i s in no
and i i )
even when
The e x p r e s s i o n
dp(Z)
will
be a u s e f u l n o t a t i o n a l d e v i c e in the t e c h n i c a l c a l c u l a t i o n s of Sections 6-11.
§3.
Analytic Ellipsoids If
domain
m = (mI . . . . . mn)
i s an n - t u p l e of p o s i t i v e i n t e g e r s , d e f i n e t h e
~ = Em t o be
2m 1 Em = ( z e C n : ] Z l l
2m n +...+lZnl
< 1}.
There i s no l o s s of g e n e r a l l t y to always suppose
that
mI S m2 S . . . S mn
and t h l s w i l l be done below without comment. Of course when m $ (1 . . . . . 1) points
m = (1 . . . . . 1)
then
(z I . . . . . Zn)
mj > 1.
then
Em i s the u n i t b a l l .
For
Em w i l l not be s t r o n g l y pseudoconvex p r e c i s e l y a t f o r which t h e r e i s a
j
satisfying
zj = 0
and
This follows from d i r e c t c a l c u l a t i o n , the d e t a i l s of which a r e
omitted.
If
mk+1 . . . . . mn
pseudoconvex p o i n t s in dimension
2n - 3;
a r e the i n d i c e s which exceed u n l t y then the weakly
8Em
c o n s i s t of the union of
n-k
r e a l e l l i p s o i d s of
each of t h e s e e l l i p s o i d s 2 n t e r s e c t s each of the o t h e r s ,
and with t r a n s v e r s a l c r o s s i n g s . If
~ CC Cn
P r o p e r t y K: K CC ~
That fact
it
If
z 0 E ~, ~j 6 Aut ~,
holds t h a t
~ = Em that
i s a domain, d e f i n e
each
Section 10.2). A closely
~ ( z ) -~ P
satisfies point
related
Property of
Em
has
and
~ j ( z 0 ) --~ P e 5~
uniformly f o r
K follows
by a standard
a holomorphie
phenomenon to
Property
then f o r any
z e K.
peaking
K is
argument function
Condition
using (see
W for
the
[22],
a domain
129
C o n d i t i o n W:
P e 5~ It
such that
z0 E £ ,
a sequence
~ j e Aut ~ ,
and a
~ j ( z O) --~ P.
is p o s s i b l e in p r i n c i p l e f o r Condition W to hold without Property K (e.g.
when z0
There i s a p o i n t
P
i s a p o i n t o f s t r o n g p s e u d o c o n c a v i t y ) . C o n v e r s e l y , t h e r e may be no
s a t i s f y i n g P r o p e r t y W, hence P r o p e r t y K would t h e n h o l d v a c u o u s l y . If
fl
i s s t r o n g l y pseudoconvex and C o n d i t i o n W h o l d s t h e n (by Bun
Wong's theorem) P E Off
fl
is the b a l l .
However C o n d i t i o n W can h o l d when
i s a weakly pseudoconvex p o i n t .
i t holds for
fl : Em
i f and o n l y i f
Indeed, a s w i l l be seen i n a moment,
mI = 1.
If all
mj > 1
c o n t a i n s o n l y r o t a t i o n s and p e r m u t a t i o n s o f v a r i a b l e s The p r o o f s o f t h e s e a s s e r t i o n s a b o u t First,
if
mI = I
~a : (Zl . . . . .
Zn) ~
i s an a u t o m o r p h i s m o f
Em
then f o r
a e C,
zl-a
(1-]a]2
[1_~Zl
Em.
then with
Aut Em mj = mk.
a r e now g i v e n . [al
)
< I,
1/2m2
Re(1-~Zl)
t h e map
(1-la[
z2
(l_~z1)1/m2
Here
zj,zk
.....
> 0
for
_
2 1/2ran )
(l-azl) z E Em
) Zn~
1/m n
|
J
so that
the
p r i n c i p a l b r a n c h o f l o g a r i t h m may be d e f i n e d . Now t h e c o l l e c t i o n o f automorphisms ~j = ~-1+1/j' the point P = (1,0 ..... 0), and ( s a y )
Zo=O
s a t i s f y c o n d i t i o n W. For t h e c o n v e r s e , n o t i c e t h a t i f on
Em w i t h some f a m i l y
then i t must be t h a t some
mI > I
and c o n d i t i o n W i s s a t i s f i e d
{~j} q Aut Em and some Pj = 0.
For o t h e r w i s e
P = (P1 . . . . . Pn) E ~Em P
would be a p o i n t o f
s t r o n g p s e u d o c o n v e x l t y and Bun Wong's theorem would imply t h a t biholomorphic to the b a l l .
Say f o r s i m p l i c i t y t h a t
Em
is
Pn = 0, P1 ~ O.
Let
S = {ze~Em : Zl=O}.
Then
S
is a real
points in
2n - 3
d i m e n s i o n a l e l l i p s o i d o f weakly pseudoconvex
5Em which i s d i s j o i n t from
P.
Now a l l automorphisms o f
c o v e r i n g t r a n s f o r m a t i o n of the b a l l in the sense t h a t i f there is a
~ E Aut B
such t h a t t h e diagram
Em - - ~ TmI
B ie commutes, w h e r e
Em tTm
¢ e Aut Em
Em a r e then
130
Thls
follows
smoothly
from results
to
generated
~E m
(see
by the
mn = 1)
(see
moves
0
will
points
in
strongly
~E m
then
satisfied
the
when
implies
that
the
at with
group
to
~a
is
map
an
strongly
such
pseudoconvex obtained
B
is
m1 = . . . of
Em
which
pseudoconvex
that
smooth on the
is
extend
of
~j
of strongly s e S
~j,
Em
(with
any automorphism
contradiction
above
proved
And b e c a u s e 1,mn)
paragraphs)
satisfied
of
automorphism
The g e n e r i c i t y
the
points
automorphisms
mappings
that
there
But then
are
m = (1,1 ..... preceding
composing
§4.
of
which
m1 = 1.
case the
type
the
mk z n ).
~j(s)
closure
points
is
of
Em,
and that
and Condition
is
W is
not
m1 > 1.
As a r e s u l t of
mI (z 1 . . . . .
and by the
follows
desired
B
Thus all Since
~ S.
Thus the when
[28]. [2]).
it
pseudoconvex
impossible.
z n) ~
~j(S)
pseudoconvex.
maps w e a k l y
(z 1 .....
group
[22]),
satisfy
: Em ~
of
also
unitary
[27],
Tm
P E ~E m
of is
then
P it
Main Technical
m
paper
It is
of
must have
in
follows this
the
may b e a s s u m e d
condition
can only
inherent
considered. if
concerning
present
limitations
that
a rotation,
The T h r e e
considerations in the
the
W, t h e o r e m s
be true proof,
(just
for only
form
(eie,0
the
as was argued
form and Condition
that
Em
.....
0).
P = {1,0 .....
0)
in
W is After =~ .
Lemmas
The p r o o f s o f T h e o r e m s 1 . 1 a n d 1 . 2 c o n s i s t In constructing blholomorphisms as limits of certain normal families of mappings. The w o r k involved is in seeing that the normal limit does not degenerate to a constant mapping. The f o l l o w i n g s i m p l e lemma i s t h e k e y t o s e e i n g t h a t a n o r m a l family of mappings does not so degenerate.
Lemma 4 . 1 :
Let
~1'
relatively
compact
then
is
there
an
~ 2 ~ Cn
subdomains N = N(E)
be domains.
For
of
the
such
~i that
with
g~ D E
i = 1,2, property for
1 be a biholomorphic mapping of a neighborhood Uj
~2
such t h a t
j
then the sequence
l i m i t mapping
if
of
be
E O= ~ i
For each
gjI
.
onto
j,
let
f j (u~) c _
2 f(K~) = Kj.
I f there is a P0 E ~I all
that
j ~ N.
fj
K i1 , K 2i , . .
let
f : ~1 - e ~2
and a s e t (fj} and
L0 (:~ ~2
such t h a t
has a subsequence f
{fjk}
is a biholomorphism.
fj(P0) E L0 wlth a normal
for
131
It in
[2 3]
s h o u l d be r e m a r k e d t h a t and have been used in
variants [9],
of this
[10],
[11],
result [12],
appear,
[13],
for
[14].
instance,
Parts
of
t h e p r o o f g i v e n i n S e c t i o n 6 may be s o m e w h a t n o v e l . The n e x t Lemma i s u s e d o n l y t o e s t a b l i s h Lemma 4 . 3 . A glance at the s t a t e m e n t o f Lemma 4 . 3 r e v e a l s t h a t i t p r e s u p p o s e s t h e e x i s t e n c e o f a p o i n t P E ~
at which
and o n l y i f
Q~(P)
every
P e fl
is not biholomorphic of 4.3 to verify Lemma 4 . 2 : Then
~
Let
> 1.
~
By Bun W o n g ' s Lemma, t h e r e
has this
property;
to the ball.
a nd t h i s
Therefore,
be a s i n t h e h y p o t h e s e s
it
is
is
one s u c h p o i n t
so if
is essential
a nd o n l y i f for
o f Theorem 1 . 1 o r Theorem 1 . 2 .
techniques
by d e f i n i t i o n , that ~ will
~9 h a s ~ as a weakly pseudoconvex point, it n o t be b i h o l o m o r p h i c t o t h e b a l l . But t h e e x i s t i n g
for establihsing
biholomoprhlc
inequivalence
of domains requires
s m o o t h e x t e n s i o n o f b i h o l o m o r p h i c maps t o t h e b o u n d a r y ( s e e , f o r [2]). And t i l e s m o o t h e x t e n s i o n r e s u l t s require that ~ itself smooth,
since
the domain
boundary,
proving
the proof
is not biholomorphic to the bali.
Since, is expected
Cn+2)
4.2.
it
~
i n Theorem 4 . 1
is necessary
See s e c t i o n
7 for
to devise
(resp.
4.2)
instance, be v e r y
has only
a new, a nd n o n - t r i v i a l ,
C3 ( r e s p . method for
the details.
B e c a u s e o f Lemma 4 . 1 , t h e p r o o f s o f T h e o r e m s 1 . 1 a nd 1 . 2 amount t o constructing a sequence of "approximate biholomorphisms" fj and p r o v i n g non-degeneracy fact,
condition
what i s proved f o r
following
result
Lemma 4 . 3
(Pseudocollared
and
if
P0 e fl"
consisting
of the existence
the situation
in other
N e i g h b o r h o o d Lemma): is
a number
P0
a nd
i n T h e o r e m s 1.1 a n d 1 . 2
(which should prove useful
Then t h e r e
of
Let
nO = n0(P0),
is
L0.
the
In
the
contexts}: ~
be a s
i n Theorem 1 . 2
a neighborhood
V
of ~,
and a s e t S = {z e fl : z = ~'+tu{., { E 8fl, satisfying (4.3.1)
the following For
no
-nll<11-11/l~nll¢11-111
< t < o} ~
condition:
¢ e Aut ~
does
it
hold that
@(P0) E S ~ V.
Lemma 4 . 2 i s u s e d t o p r o v e Lemma 4 . 3 . Lemma 4 . 3 i s t h e k e y t o v e r i f y i n g t h e h y p o t h e s i s o f Lemma 4 . 1 , and i n t h e p r o o f o f Lemma 4 . 3 l i e s m o s t o f t h e t e c h n i c a l wo rk i n t h i s p a p e r . In the next section, assuming the correctness o f Lemmas 4 . 1 , 4 . 2 a n d 4 . 3 , t h e p r o o f s o f T h e o r e m s 1 . 1 a n d 1 . 2 a r e g i v e n . Lemmas 4 . 1 a n d 4 . 2 a r e p r o v e d i n S e c t i o n s 6 a n d q r e s p e c t i v e l y . The p r o o f o f 4 . 3 i s c o n t a i n e d i n S e c t i o n s 8 t h r o u g h 13.
132
§5.
The Proofs of Theorems i.I and 1.2 (assuming Lemmas 4.1, 4.2, 4.3) First Theorem 1.1 will be considered. Fix m, E m, ~, and p as in the statement of Theorem i.I.
supposed that there is a d O > 0
It may be
such that
0 fl B ~ , 2 d o )
(5.1)
~ Em ~ B ( t , 2 d o ) .
From now on, (5.1) is assumed and the subscript p is omitted. Let PO = (1-do'O) e ~ and let S~ be the corresponding pseudoeollared neighborhood for SEa
~
and
V = B(1,2do)
given by Lemma 4.3.
be the pseudocollared neighborhood for
S~ = {z • fl : z = t + t u ¢ ,
corresponding to
PO"
-~111{'11-11/1£n[~¢11-111
< t < O}
SEre = {z e Ea : z = ¢+tu~, { e ~Em, ~211Cll-ll/l~n{[~1{-lfl
< t < o)
Let
wO
Em
and
be the minimum of SQ
is replaced by
{" e 3~,
~E m
Likewise, let
~1
and
W2"
S~ ~ SEm n ~
If
SEm
is replaced by
Then
SEm ~ S~ n
then the conclusion of Lemma 4.3
will be satisfied and it will further hold that
n B ~ , R d O) n SEm = ~ n B(~ 2d o) fl Se. Replace
{~j}
by a s u b s e q u e n c e
if necessary
~ j ( P o ) m Pj = ( a j , b j ) Choose
oj E Aut E m
such that, setting
with
to achieve
Ii-ajl
(5.2)
the condition
< do .
hj = ojo~j,
hi(P0) m oj o ~j(P0) m p~ m (a~.b~) satisfies
a S = 1 - dO
(refer to Section 3 for details about
Notice that, by construction,
B(~,2do)
(line
(5.2)).
Pj ~ S~ n B(~,2d0)
By Leama 4 . 3 ,
hence
Aut Ea). Pj ~ SEa
133
P~ @SEmfl B(~,2do). It
(5.3)
follows that P) e Em fl ( ( 1 - d o , z 2 )
: dist(z,OEm)
k 1 - nO(1-do)/l£nll-doll}.
As a result, n 0 m (p~} ~ In order remains
to apply
Lemma 4 . 1 w i t h the existence
to verify
Since
Tm : Em--~ B
the Kobayashi metric,
PO = ( 1 - d o ' O ) of
{K~},
Em
(5.4) as above and
it
L0 = ( P j } ,
(K~}.
is a holomorphic
then
Em.
covering
is also complete.
gm K~ : {z E E m : d i S t K o b ( z , O )
and
B
is complete
in
Let
< £}.
Then
...
K12 CC K i CC
O : Em
and
t=l If
j = J(£)
is
large
K{
=
Em.
then
oil (K~) cc E~, fl B(~,2d 0) and,
necessarily,
= (~
o ;j 1 ( K ~5 By 6 r a h a m ' s function at
Em gm : distKo b (z,~j(Pj))
localizatlon arguments in [8], ~ now g u a r a n t e e s that if j
< £).
the existence of a local peaking is sufficiently large then
E~B(~,2d O)(z,¢j(PO )) aj1(_t)=K~ 2 {Z e g m : d i s t K o b < £ - 1}. Since
the
inclusion
map i
: fl fl B ( ~ , 2 d o )
--+ Em fl B ~ , 2 d O)
134 is distance
decreasing
in t h e K o b a y a s h i m e t r i c ,
it
follows that
~B(~'2dO)(z,~j(Po)) ~31(~)=K? ~ (Z ~ fl : diStKo b
Again by t h e l o c a l i z a t i o n
of the Kobayashi metric
it
o;l(Kff)j~ ~ {Z E ~ : disVQ~ob[~ (Z,~,(Po))j
- 1}.
< £
follows that < £ - 2:}.
Finally, s e t
(~j (£) o
j(£)
Then
K~ a {z ~ ~ : d i S t ~ o b ( Z , P o ) < £ - 2}. Clearly oo
£ ]
Since each
that
(K~)
a -1
is relatively compact in
B(~,2d0) N B i t follows
K~ ~Y~. Passing to the subsequence
~j ~ hj(£),
h y p o t h e s e s o f Lemma 4.1 a r e s a t i s f i e d . s u b s e q u e n c e w i t h a normal l i m i t
The proof for
n > 2
h
one s e e s t h a t
Thus t h e m a p p i n g s
all ~j
the have a
which i s a b i h o l o m o r p h t s m o f
is similar:
the t r a n s i t i v i t y of
~
with
Aut Em on
normal directions together with the pseudo-collared neighborhood (for tangential directions) prevents degeneration of the normal family.
§6.
The P r o o f o f Lemma 4 . 1 Since
{fjk)~=l
~2
i s bounded,
g (fj}j=l
it
has a limit
Is trivial f
that
such that
f : fll - ~ 32 It is n e c e s s a r y to check that in fact f
is onto.
some s u b s e q u e n c e
'
f : i}l --~ ~}2'
f
is one-to-one,
and
Em •
135
m I
First
notice
that
{fj~}
has a subsequence with a normal limit
QO = f ( P o ) ~ L L - ~ .
Since
numbers
such that
r1
~ 2 ( Q o , r 2)
and
It follows
are hyperbolic,
there
the Kobayashi metric
have compact c l o s u r e s
r = min{rl,r2}. id)
r2
Ql,~2
in
~ 1 ' ~2
Let
g.
are positive
balls
respectively.
(by l o o k i n g a t t h e n o r m a l l i m i t
p
(Po,rl)
and
Let of
f J k o f-ljk =
that
f o g = id
on
fl
(Qo,r2).
(4.1.1)
Therefore det f' i s not i d e n t i c a l l y zero. Now observe t h a t det f ' , the normal l i m i t of the non-vanlshing functions det f~, i s e i t h e r
being
i d e n t i c a l l y zero or non-vanishing. Since det f' i s not i d e n t i c a l l y zero, i t must be t h a t det f' i s never zero. Hence f i s an open mapping and
f(~l ) ~ Q2'
To check the univalence of f , l e t B(O,R) be a Euclidean ball which contains ~i" I f z,w E ~ are fixed d i s t i n c t points and i f j i s so large that
Kj 1
{ z , w } then i t holds t h a t
K1. K2 distc3ar ( f j ( z ) , f j ( w ) ) = distc3ar (z,w) --
.B(O R)cz w~
-> (11S~car'
By t h e u p p e r s e m i - c o n t i n u i t y follows that
of the Cartheodory metric
diStcar(f(z)
Reversing one.
Finally,
f o g E
id
the r o l e s (6.1.1)
on all
of
of
the proof.
§7.
f
yields ~2'
~ ' " --- ~ z , w
'
and that
Likewise,
g
yields
g o
[25],
[22]),
it
> 0"
f(w)) > ~z , w -
f o g = id
(see
> 0.
that
g
: ~2 -~ ~I
on an o p e n
f E id
on
~I'
set; This
is o n e - t o -
hence completes Q
The P r o o f o f Lemma 4 . 2
Since this result is particularly e l u s i v e , two p r o o f s w i l l be p r e s e n t e d : the first in detail, t h e s e c o n d i n o u t i n e form. The f i r s t p r o o f u s e s localization o f t h e C a r a t h e o d a r y and E i s e n m a n volume f o r m s . The s e c o n d p r o o f u s e s an i n t e r e s t i n g a p p l i c a t i o n o f Lemma 4 . 1 . F o r t h e f i r s t p r o o f , n o t i c e t h a t i t f o l l o w s from r e s u l t s o f B e l l [ 2 ] , o r
136
even from more elementary considerations [4], that
Em i s not biholomorphlc
to the ball.
Let
maps of
In p a r t i c u l a r ,
Em such that
QEm(#j(0)) : i + ~0"
QEm(0) m I + t 0 > I.
~j(0) = ( l - i / j , 0 . . . . . 0) --~|_. Let
a0
#j
be biholomorphic
Then of course
be a small positive number. Then the usual
localization arguments for M C and M E (see [8] or [22], for instance; the delicate quantitative version of localization in Section 11 i s not needed here) show that
j
QE~B(~,a0)(#j(0)) ~ i + 3~0/4 for
large.
Now
elementary comparisons, using of course the hypothesis ( i i ) in Theorems 1.1 and 1.2 , shows that
0C~B(1,a0)(#j(0))
Finally, localization for
~ I +
Q implies that
~0/2
if
0~(#j(0)
j
is large.
~ I + ~0/4
if
j
is
large. Thus 0 i s not biholomorphic to the b a l l . A second proof of Lemma 4.2 proceeds as follows I f ~ were biholomorphic to the b a l l , then i t would have a t r a n s i t i v e group of blholomorphie s e l f maps. As in the beginning of the proof of Theorem 1.1 in Section 5, replace ~ by
Q# = {z : p(Zl,(l+#)z 2) < 0 } (for a small real
~)
so that for some small
dO > 0
i t holds that
n B(~,2d O) ~ Em ~ B(~,2do)For
j
s u f f i c i e n t l y large, ~j(P0)
Write ~j(P0 ) : ( a j , b j ) . ~j E Aut ~lu choose
such that
E ~U fl a(~,2d0) ~ Em fl B(l,2do). Since
Aut ~#
i s t r a n s i t i v e , there i s an element
~j(~j(Po ) ) = (aj,O) m Oj E ~ # n S(~,2d0).
Xj ~ Aut Em such that
Then the maps hj ~ xj o ~j o ~j
are biholomorphic mappings which
s a t i s f y the non-degeneracy conditions of Lemma 4.1 (with The existence of
Kj, i Uj i
and 1.2 in Section 5.
Finally,
Xj(Qj) = 0.
P0=P0 and Lo={O}).
i s established j u s t as in the proof of Theorems 1.1
Therefore the mappings { h i } s a t i s f y the hypotheses of
Lemma 4.1 and there i s a subsequence converging normally to a biholomorphism of
~#
to
Em-
In conclusion, [4]).
§8.
B ~ 9 ~ ~# ~ Em.
This contradiction
Uniform Estimates
establishes
for the
a
But i t
is well-known that
Lemma 4 . 2 .
Equation
B ~ Em
(see
137
In t h i s s e c t i o n i s c o n t a i n e d a v e r y d e t a i l e d form o f t h e uniform estimates for the
~
e q u a t i o n on
B e a t r o u s and Range [ 1 ] .
~.
The r e s u l t i s e s s e n t i a l l y due t o
The c o n t r i b u t i o n h e r e i s t h e more r e f i n e d e s t i m a t e s
on t h e norm o f t h e r i g h t i n v e r s e f o r t h e
o p e r a t o r which i s c o n s t r u c t e d .
Lemma 8 . 1 :
Let
kI = kl(n)
such t h a t t h e f o l l o w i n g i s t r u e .
Let P E D~ Cn+1
0 < n E Z.
~
O CC: Cn and l e t
Then t h e r e a r e p o s i t i v e i n t e g e r s
be a pseudoconvex domain w i t h U1GU U2 ¢~ Cn
defining function for
~
If
with
]Vp[ = I
on
is a ~-closed
(0,I)
form on
c o e f f i c i e n t s which a r e s u p p o r t e d i n on
D~.
~
t o the e q u a t i o n
~u = f
fl
P.
p
~
be a
t o be t h e
Assume t h a t
Let
and
Let
Let
Define
U2 n 8~.
d = diam (U1 n fl), 6 = d i s t ( u 1 , C u 2 ) , D = diam ~. f
boundary.
be open n e l g h b o r h o o d s o f
l e a s t e l g e n v a l u e o f the Levi form a t p o i n t s o f Let
CN+I
CI = Cl(n)
x > 0.
S = Hpllcn+l(cn).
which has smooth, bounded
Ul
t h e n t h e r e i s a bounded s o l u t i o n
u
with
k1 IlUllLO~(~) < C 1 • ( X ' )
IlfIILOO(a)
and k1 HU][LOO(CI\Ua) < C 1
(X')
HfIIL¢~(I~ )
where ~(' In what follows, be s t r e s s e d , U1
and
however,
solution that
will
+ 6-1
be d e n o t e d
the operator
it
should
The m a i n l i n e s
be n o t e d
U ¢ Ca
that
of the proof
d e p e n d e n c e on t h e p a r a m e t e r s . Now t h e s a l i e n t feature an o p e n
u
+ R-1
T
depends
+ S. by
Tf
or
not only
T~f. on
It
fl,
should
b u t on
U2.
Finally, Proof:
this
= n + D + d-1
such that
of the
the are
in
proof
U 1 ~-C U C~= U2
operator [1]. in
T
is
linear.
What i s new h e r e
[1] m u s t be r e v i e w e d .
and
dist(Ul,CU ) ~ dist(u,Cu2) ~ 21 d i s t ( U l , C U 2 )
is
the First
fix
138
Now, u s i n g
the
perturbation
"bumping technique" ~
of
fl
such that
properties hold: Whenever f bounded coefficients supported
(i)
a ~-closed
(0,1)
supported
of gerzman ([17]),
in
U~ ~ ~ ~
one obtains
and the following
is a ~-closed (0.1) f o r m on i n U1, then there are
form
~ ~ U
~
and
with smooth, C~
a critical
~
with smooth,
bounded coefficients
on
and
(11)
such that
a function
u 1 ~ C~(fl) D C(D)
T
satisfy
and
u1
f = T + ~u 1
(a)
I1~11L¢¢(~) _< CoIIfllL¢~(l));
(b)
HUllIL~(Q) s Co[IfllLOO(Q) "
Now
7,
uI
are constructed
[17] a n d t h e r e f o r e n, D, d - 1 ,
6 -l,
using
the constant
k-I
with support
CO
on
~
the local
in
U
and
solution
operator
of Kerzman
is well-known to depend polynomially
on
S.
I t i s known ( s e e [ 1 5 ] ) t h a t l o c a l s o l u t l o n o p e r a t o r s c a n be c o n s t r u c t e d u s i n g j u s t two d e r i v a t i v e s of the boundary. Kerzman [17] u s e s j u s t t h r e e derivatives, which suffices for the application here. For t h e n e x t s t e p , perturbation ~.
~*
of
one c o n s t r u c t s
~
such that
~*\U 2
The B o c h n e r - M a r t l n e l l l - K o p p e l m a n
write
~ = g + ~u 2
a small
Cn + l
= ~\U 2,
formula
pseudoconvex
~ ~ U ~ ~*,
{[21],
[22])
and
~* ~ U c
l s u s e d on
~
to
where
g = SO~ ~ A K1, u2 = S~ ~ A KO,
and
g O, K1
uniformly
are the usual
integrable,
Bochner-Martinelli
kernels.
Since
Ko(Z,.)
one c h e c k s t h a t
IlU2NL~(~ ) -< COOIITIIL¢~(~) < COO • CoHf]~L~(~ )
Also
the support
condition
on
T
makes
it s t r a i g h t f o r w a r d
to check
that
is
139
u 2 E C(~). CO0
The e x p l i c i t l y
known form o f
d e p e n d s p o l y n o m l a l l y on The s u p p o r t c o n d i t i o n
under the integral
sign)
K0
and
K1
make i t
immediate that
n, D, d - 1 , 6 - 1 .
on
~
makes i t
g e r~n( + 0 ,l 1 )
that
Ilgll,n+l
~,~
e a s y to check (~*)
(by d i f f e r e n t i a t i o n
and
Co0011"fllL¢o(~/)
l
~ ( 0 . 1 ) ~.. ,
As u s u a l ,
Co0 0
Now g o h n ' s provides a map
d e p e n d s p o l y n o m i a l l y on global
solution
operator
n, D, d - 1 ,
6 -1 .
developed in
[19,
Theorem 3 . 1 9 ]
cn+l An+1 : ~ ( 0 , i ) ( ~ * ) N ker
---, c°(X*) ~ c~m*) such t h a t
BAn+1 = id
and
llAn+l °'llwn+l (~,,~) _< c ' llHlwn+l (o, I ) (Q* '~) Here
~
i s a weight function of the form
corresponding weighted Sobolev c l a s s . W n+l
e s t i m a t e s depend on
be checked d i r e c t l y ) . depending on
n
and
(n+l)
e- t l z l 2
and
W n+l
the
(Kohn does not s t a t e e x p l i c i t l y t h a t
d e r i v a t i o n s of
~,
but t h i s i s so and may
The choice of weight i s fixed once and for a l l llpllcn+l.
The weighted Sobolev space i s the usual
Sobolev space with a d i f f e r e n t , but comparable norm. The l a s t l i n e , t o g e t h e r wlth the Sobolev ImbeddingTheorem ([16, p.123]) now y i e l d s t h a t
IIAn+IHIL~(~*) < c II II_n+l c(0,i ) (n*) Now g
i s P-closed because of the equation
T = g + ~u2
makes sense and
IIAn+l~lLoo(9,) S c"Coooll'rll ..¢¢(~/) < c..Coo o CoIIflIL~(~ ). Therefore Tf • An+ig + u2 + u I
so
An+ig
140
satisfies
§9.
all
the
Analysis
necessary
of Peaking
In a general
estimates
and
~Tf = f.
Functions
function
algebra,
peaking
functions
play
the
cutoff functions play in the algebra C~(M), M a m a n i f o l d . work, the abstract properties of peaking functions will not necessary to know something about their decay away from the Further, it is larger domain. Lemma 9.1:
and
required that the This information
Let
0 < n E Z.
k 2 = k2(n) Let
P E B~
such
90Z
Cn
and let defining
least
eigenvalue
of
the
d = diam(Ul~),
there
is
is
Levi
form at
with
l~pl
= 1
points
of
6 = dist(Ul,CU2), 9*
of
holomorphic
~p(P)
(9.1.2)
1/2 ~ I ~ p ( z ) l
(9.1.3)
I~p(z)l
Cn + l
9
on
such
to a
C 2 = C2(n)
that on
boundary. of
P.
8~.
p X
be a to
Assume that
such
be the ~ > O.
S = Hpllcn+l(Cn).
~*\U 2 = 9\U 2 9*
Let
Let
Define
U2 ~ 5 9 .
D = diam 9,
~p
(9.1.1)
integers
is
true.
domain with
9
a function
is
be open neighborhoods
for
a perturbation
and there
there are p o s i t i v e
following
be a pseudoconvex
function
that
peaking functions continue analytically is contained in the following lemma.
Then
the
UI CC U2 ¢ C n
Cn + l
Let
that
role
In the present suffice and it peak point.
Then
a n d ~ ~ U1 ¢
that
= 1 < 1
S 1 -~
for
z e n U (a9flUl)\{P}
• ~(z,P)
where -k 2
c2(~(')
and at
X' the
Proof:
= n + 6 -1 + d -I end of Section This
is
the
k2
_< ~ < c2(•'
)
+ x 1 + D + S
and
p
is
standard
construction
of a peaking
pseudoconvex point which can be found in [22]. locally as the exponentiated Levi polynomial. then
hold
globally derivatives
locally
by inspection.
by using
a cutoff
of
cutoff
the
the
skew distance
defined
2.
function function
The f u n c t i o n The e s t i m a t e s
The p e a k i n g
function
and solving introduce
function
is
a suitable
poiynomlal
at
Is constructed {9.1.1)-(9.1.3)
then ~
defined problem.
dependence
and the Levi polynomlal and estimates for the ~ problem introduce polynomial dependence on the other parameters.
a strongly
on
i n Lemma 8 . 1
The 6 -I ,
141
F i n a l l y the r e s u l t i n g peaking function to guarantee t h a t (9.1.2) holds g l o b a l l y .
§10.
Uniform
Approximation
Proposition I 0 . i : P • ~
and l e t
Let
of
Holomorphic
~ CC Cn
Let
UI CC U2 CC Cn
~p
replaced
be neighborhoods of 3~
by
Cn+l P.
are bounded from
boundary.
0 < a < b < 1
U2 = {z e ~:n : [ @ p ( Z ) f
Let
Assume t h a t the
0
by
k > 0
be the peaking functin given by Lemma 9.1.
there are numbers
~ ~ (¢+3)/4
Functions
be pseudoconvex with
eigenvalues of the Levi form on U2 ~ ~ '
is
on
Suppose t h a t
such t h a t
> a),
U 1 = {z e ~ n
: ]~p(Z)[
> b}.
Let S, d, g, D be as in Lemmas 8.1, 9.1 and l e t ~ > 0 and 0 S k E Z. Then there i s a constant L such t h a t i f f is ~ bounded holomorphlc functions on
U2 n ~
then there i s a bounded holomorphic
T
on
~
such
that
(10.1.1)
for
,
sup ZeUl[D
all
-
SUCh t h a t
multi-indices
II}II L~(~)
(10.1.2)
:
al S k,
and
< L " llfllL~(U2)
Here
L ~ {(C3x')kl/(~ck)}
where
~'
= n + d - 1 + 6 -1 + X-1 + D + s
and
4/(b-a)
C3 ,
c,
k1
are
positive
constants. Proof:
This
follows
loss
of
generality,
Let
U~ = (z e ~
the
ideas
that : l~p(Z)l
in
[8,
b > a > 3/4. > a'},
p.230]. Set
It a'
U~ = (z e dom ~ p
so that
(i)
(ll)
v 1 oc:
~\v~ = ~\v~
will
be supposed,
= (3a+b)/4, : l~p(Z)l
b'
without
= (a+3b)/4.
> b'}.
Select
142
(ill)
Let
~I = (~\~) U ( U ~ ) .
0 ~ ~ S 1,
~ = I
{~p{ t b'
~\~.
Define a cutoff function
on ~ ~ U~, ~ = 0
= dist(u1,Cu2).
on
Choose c > 0
~ • C~(O) such t h a t
on O\U2, ]V~{ ~ ~36-I,
where
such t h a t
d l s t ( U 1 , Q \ U ~ ) ~ c5,
dlst(Ul,~1)
Notice t h a t
c, ~3
~ c6 .
bay be chosen to depend polynomiaily on the usual
parameters. Now g i v e n
f E H~(U~U),
g(z)
define
= I {~(z))f(z)
Lo Then
g E L~0,1) ( ~ ) ,
hence
I~I ~ a'
on
g
i s s m o o t h , and
s u p p g.
T
i s the
8
.
II}-fll
T
D
L~(U1 )
if
z E ~ \ U 2.
~g = 0.
Notice that
f
g = 0
on
U2, ' .
_ ,;r
s o l u t i o n operator of Lemma 8.1 and
selected. Direct c a l c u l a t i o n shows t h a t the operator
z E U2 fl
Now l e t
} °
where
if
shows t h a t
f
0 < r E Z will be
Is holomorphic and Lemma 8.1 concerning
f E H~(~).
Furthermore,
= H3~-rT (@rg)IiLoo(~1) _< C1(~' )kl(b' )-rn~rgJiL~(~ ) _< CI(~' ) k l ( a ' / b ' )-ri{gllLOO(O) _< CI~3~-1(~ ' ) k l ( a ' / b ' )-rIlflI oo
L (U2{I0) Here i t
is understood that
~'
is the constant
X'
from Lemma 8 . 1 computed
143
relative to {7, U 1, U~.
Now
r = where
[ ]
let
in ~ ek ~ ck
j/tn(b'/a')
is the greatest integer function.
apply to the holomorphic z • U 1 ~ ~ CX: U 1
and
function
i~[ = k
+ 1
Then the Cauchy estimates
f - f = ~-rs(~rf)
on
~1"
For
one thus obtains
II[~z)(f-f)ltL~(%m) _< k~ (c5) -k IIf-fllL~(l~l ) < kf ( c s ) - k C l ( ~ ( ' ) k c 3 d - l ( a ' / b '
)roifHL~(U2[~I)
< ell fllr®(u~) by the choice of
r.
Also
^
I[fliLoo(~}) -< JlflJLco(U2l~) + H~-rT (y~rg)H L~ (1)) -< II fll Loo(U2~ )
+ 2riDT- (~rg)ll oo
~'/
-< ][filL= ( U 2 ~ ) + 2rc1(~ ') -< ][fIIL~(U2~ )
L (9) kl
r
ll~h gHLoO(• )
+ 2rCl(~' )kl(a, )rc35-11lf[ILo~(U2~)
This last is obtained by recalling the definition of g, and the e s t i m a t e on Iv~I. This is
f
k
12°.
k!C I ( ~ ' ) 1U3 -<' 1 + L~fk+l e c k J
by the d e f i n i t i o n
of
r,
~skck m~
the support of
IIfiiL°°(U2[~) '
g,
144
-
k
_< l+t k+i ck j I llfll L'(U~)' J B
" LH fll L~, ( U 2 ~ )
§ii.
L o c a l i z a t i o n of the Caratheodary and Eisenman Volume Forms In order to use
O~
in l a t e r normal f a m i l l e s arguments,
now to compute the dependence of
M~ and
~
i t i s necessary
on l o c a l d i f f e r e n t i a l boundary
data. This computation n e c e s s i t a t e s e x p l i c i t formulation of c e r t a i n l o c a l i z a t i o n arguments. The l o c a l i z a t i o n arguments for the metrics under c o n s i d e r a t i o n here were pioneered by Graham in [8]. His work was done in a C2 s t r i c t l y pseudoconvex domain, and h i s l o c a l i z a t i o n e s t i m a t e s are t h e r e f o r e uniform over points of the boundary. By c o n t r a s t i t i s necessary in the present paper to l o c a l i z e the metrics a t s t r o n g l y pseudoconvex p o i n t s which are very near to a weakly pseudoconvex point: the e s t i m a t e s change?
as the Levl form degenerates, how do
The e s t i m a t e s obtained below should have broad a p p l i c a b i l i t y to the function theory on domains in which s t r o n g l y pseudoconvex boundary p o i n t s are generic. The t h r u s t of the c a l c u l a t i o n s l s to see t h a t no global information about the domain except for i t s diameter plays a r o l e in the l o c a l i z a t i o n arguments.
Proposition P e 5~
II.~:
Let
~ C C En
and suppose that
U2 c Cn
elgenvalues of the Levl form on U2 ~ 5Q.
be p s e u d o c o n v e x with
Suppose further that
C n+l
is a n e i g h b o r h o o d of 5Q
U2
are bounded from
P 0
boundary.
Let
such that the by
x > 0
on
has the form
U2 = {Z : l#p(Z)l > a} where
#p
is the p e a k i n g function of Lemma 9.1.
a constant w e U~
b', a < b' < 1,
such that if
C
Here
S
~4]u2(w)
~(w)
~ > O.
U~ = {z : l#p(Z)l
then
I
Let
S 1 +~.
Then there is > b'}
and if
145
b'
c'
= 1 - c'
• ~ •
i s a small p o s i t i v e c o n s t a n t , and
Proof:
Let
function
~ > 0
be such
F e B(C4]U2)
(this can be done since
that
~),
~ = n + D + -1
(l+~}n- = 1 + e. (l-v) n
such that B
(1-a)2/(ln
F(w) = 0
and
Let
Then
an n-tuple of complex valued, bounded functions. Set b = (a+2)/3 and U 1 = {z : l#p(Z)l > b}. ~,
replaced by
~/n.
U I, U2
One therefore
w E Ui.
[det F' (w){
is taut -- see [22]).
I0.I may thus be applied to
+ X-1 + S + ( l - a ) - I .
and each
Select
c = ~4]u2(W)
F = (F 1 ..... F n)
The approximation
Fj
obtains a function
with
a
m=l
~ : ~ __~ ~n
is
result
and such that
^
llF-FIIL~(UI[]Q ) < n
{{~{{LOo (0) ~ L = F(w) = 0
F(w)
F'(w)
=
F' (w).
(The l a s t two conditions are arranged, of course, by s u b t r a c t i n g o f f a l i n e a r polynomial w i t h small c o e f f i c i e n t s .
The c o n s t a n t
L
is s p e c i f i e d in I 0 . 1 . )
Set
In [ 1 0
60 :
where
}'
is
the constant
(l-a)
X'
C1(~')
L -J 7
2
from Lemma ] 0 . i computed r e l a t i v e t o
the s e t s u I = (z
:l~p(Z){
> (a+2)/3}
and O2 = ( z
: I~p(z)l
> a}.
Define
Y(z)
Notice
that
= (#p(Z)) #
• F(z).
Q
and
146
and
;'(w) = ~(w)F'(w) =
yPpp(W)F'(w).
Therefore
Idet Y'(w) I ~ 14(w)l n = by t h e
choice o f
F.
i~(w)l
n
Notice that since
Idet F'(w) i
•
c MUzFID(w)
w 6 U~
it holds that
l@(w)l ~ b'
so
14(w)i ~ b'ft. The c h o i c e
of
b'
and
and a l i t t l e calculatlon then reveal that
#
lY~pp(W)i ~ (I-7). Thus
Idet
P'(W) I £ (1-~)nM~2~(w).
On the other hand, by the approximation
result,
II~IIL~(~\U~) S bPL S 1. The v a l u e direct
of
L
calculation.
is
given
in the
proof
of
10.1
and the
estimate
Also
HYlIL~(UIFp ) -< IfFfl C ( u { ~ )
+ n
< 1+~?.
So
~
• Y ~ G e B(D)
and
M~(w) ~ idet G'(w)l ~ (1-r/)n (~+v)n I ~ 1
u2(W)
c
=7T~ " MEau2(W)"
follows
by
147
Proposition Then there
w E U~
Let hypotheses be as in Proposition i i . i .
11.2: is
a
b", a < b " < 1,
such t h a t i f
Let
e > O.
U~ = {z : l#p(Z)l > b"}
and
then
I
I_<
Indeed,
the
ME(w)
< 1 +~.
choice
b" = 1 - c"e4n(l-a) 4n w i l l do, Proof:
c"
a small p o s i t i v e constant. w E ~ fl U~.
Fix
Let
P e fl(B),
F(O)
: w.
Consider
~p
o F.
Notice
that
_ [~poF(O) I _ b" < < ~n ~
where
l~poF(¢)Id v o l ( ¢ ) , B
i s the Euclidean volume of the b a l l
An
1-(c,)l/2~2n(1-a) 2n.
Since
suPi@poFI S I,
B g Cn.
Set
~,, o
i t follows t h a t
vol{zeB : l~poF[ > b"} ~ b" • An .
Set
W = ((
E fl : l@p($)l
> b"}
and E = {z ~ B
Then
0 E E
B(O,l-r) (since
and
is the
vol(E)
within
volume of
~ b " A n.
Euclidean E
is
[~p(V)[
: F(Z)
Let
y = ~/2n.
distance so
E W}.
(1-b") 1/2n
large).
Thus,
~ [~p(e)[
(where e E E i s an element s a t i s f y l n g i s , by the Cauchy e s t i m a t e s ,
that
every
of an element
point
of
of
E
v e B(O,l-r),
- [~p(e)-'~p(V)[
Ie-vl S ( l - b " ) I / 2 n )
b" - 2(1-b")l/2n T
for
Notice
s b"'.
and
this
last
148
Consider
the
function
z --~ F ( ( 1 - v ) z ) .
Notice t h a t
~(0) = F ( O ) = w.
It follows
~41~'(w) ~
Since
this
follows
inequality
that
1
I
1
Idet Y'(O) I
l(1-~)n[
Idet F'(O)I
holds
for
any
F e ~(B)
such
that
F(O)
= w,
it
that
I(1-~)nl
< (l+~)~(W). It remains
For
or
this,
that
it
is
11.3:
peaking
that
enough
if" ~ a .
Corollary be the
to c h e c k
to
This Let
see
last fl,
function
that
assertion
P
be a s P
at
follows
in
which
the is
by an easy
previous provided
calculation.
two p r o p o s i t i o n s . by Proposition
9.1.
D Let
Suppose
that
u 2 = {z : l ~ p ( z ) l is
a neighborhood
3 ~ ~ U2
are
of
bounded
P
such
from
0
that by
the
> a}
eigenvalues
x > O.
Let
of
~ > 0
b 0 = 1 - C4 • ~ 4 n ( 1 - a ) 4 n / ( l n
(with
X = n + D + -1
+ k-I
+ S + (l-a) -I
and
: h~p(Z)l
Levi
and
let
forms
on
X)
C4
Define
U0 = {z ~ C n
the
> bo).
a suitable
#p
constant).
149
If
w e U0 ~ ~
then
(l-~}0~(w) ~ 0c4qu2(W)s (l+~)0~(w). Proof:
This is immediate from the two propositions.
It is the Corollary which will be used in what follows to estimate
§12.
E s t i m a t e s on t h e Biholomorphic I n v a r l a n t
%(w).
Q~
The purpose o f t h i s s e c t i o n i s t o prove t h e c r u c i a l e s t i m a t e s on which w l l l l e a d t o Lemma 4 . 3 .
Lemma 1 2 . 1 :
Let
(~
F i r s t a t e c h n i c a l lemma i s needed.
Q : cn × ~n__~ C
be a positive
definite
quadratic
form.
Let
DQ = {z e Cn
: Re z 1 < - Q ( z , z ) } .
Then t h e r e i s a b i h o l o m o r p h i c map
Proof:
(see
[8]).
Write
one can diagonalize
~
Q(w,w)
= aljwiw j.
Since
Q
is positive definite,
n
the
i,j=2
aijwiw j.
So
DO
i s biholomorphic to a r e g i o n of
form n
n
Re w 1 < - Z I ~ j w j l 2 - 2 Re Z a j l w j w 1 - a l l ] W 1 1 2 . j =2 j =2 A change o f v a r i a b l e o f t h e form
Z 1 = w1 zj
= wj + fljW 1 ,
j = 2 .....
n,
e l i m i n a t e s t h e c r o s s terms and y i e l d s
]ao(zl_fll)[2
n
2
j=2 F i n a l l y , t r a n s l a t i o n and d i l a t i o n o f z 2 . . . . . zn ,
give the region
z I,
t o g e t h e r with d i l a t i o n s of
150
n
Iz~l J
Z
j=l Notice that the biholomorphlsm
2 < I.
of
DQ
to
B
is not unique,
applications of this lemma in what follows will, for convenience, particular blholomorphism ~Q constructed in the proof.
Proposition P E 5~
12.2:
be a strongly
function. the
Levi
Let
Let
Let
be pseudoconvex
pseudoconvex
U2 = {z
form of
3 ~ N U2"
~ CC C n
~
: I~p(Z)l
has
~ > O.
point
eignevalues
There
is
and
> a}
with
~p
the
Cn + l
bounded
a number
from
use the
boundary.
associated
be a neighborhood 0
of
by
Let
peaking P
such
X > 0
b O, a < b 0 < 1 ,
but
such
that
on that
if
is
P
U0 = {z : ~p(Z) > bo}
and i f
w E U0 N fi
and t h e o r t h o g o n a l p r o j e c t i o n of
I < Q~(w)
The number
b0
w
to
~fi
then
< I + ~.
may be taken to be
b0 = 1 - C5(1-a)4ne4n/(ln(~())
where
X = n + D + -1 Proof:
+ k- ]
+ S + (l-a) -I
By a s t a n d a r d c a l c u l a t i o n ( s e e [22, Lemma 3 . 2 3 ] ) , t h e r e i s a
neighborhood
W of
P
and a l o c a l change o f c o o r d i n a t e s on
W such
that
becomes t h e o r i g i n and p ( z ) = Re z I + Lp(Z) + N(z)
is a defining function for L e v i form and
fl
in
N(z) = O ( I z 1 3 ) .
W.
Here t h e q u a d r a t i c form
Suppose i n advance t h a t
now on work i n t h e new c o o r d i n a t e s .
Set
~ = ~X/IO
U2 ~ W,
and d e f i n e
E- = E~(P) = {z : Re z I + Lp(Z) + ~ I z ] 2 < 0}, E+ = E : ( P ) = {z : Re z I + Lp(Z) - ~ I z l 2
There a r e c o n s t a n t s
P2 < ~1 < 1
such t h a t i f
< 0}.
Lp
is the
and from
151
v 1 = (z : ISp~z)l
> ~ 1 },
v 2 = (z : I ~ p ( Z ) I > p 2 ) , then
E- fl Vl c ~ fl V 2 ¢ E +. Of c o u r s e
the choice
of
# 1 ' P2
d e p e n d on
~hp
and polynomially
on
k -1 , IIPllC2. According to Corollary that
if
11.3,
w e C0 -= {z : I S p ( Z ) l
there
> bo}
(1-~/3)Qfl(w)
in
b O,
it
[1
also
-
w 6 i}
b o ' ~1 < bo < 1,
such
then
< QflflVl(W) < ( l + ~ / 3 ) O e ( w ) .
A c h e c k o f t h e form o f t h e c o n s t a n t adjustment
is a number and
holds
_(w) ~]M E
b0
shows that,
with a negligible
that
< M
(w) <_ (1 + £ M 31
E-flY1
E_(w)
and
Here
M
can be taken
to be
MC
or
ME.
N o w for
w e U0
1 _~ O~(w) = ~ ( w )
<
E ~V
2 (w)
-
[ 1(w)
ME
<
E -~Vl E+~[V1
-
it
holds
that
152
E
where
~
i s the Jacobian d e t e r m i n a n t of the biholomorphism ¢ : E+ -~ E-
(provided by Lemma 12.1) e v a l u a t e d a t 181 ~ I - ce,
Replacing
§13.
some e > 0.
~
Proof
by
of
e/(3(c+l))
~
such
completes
~
that
is
not
OQ(po)
c Then
¢(C
) = C
e Aut ~. contains
~ E n, eggs
for
t
every
T h e Lemma w i l l
simplicity n > 2,
the
is
the just
holomorphic
proof.
to
1 + 2e > O.
= {z e e
a pseudo-collared
For
An easy c a l c u l a t i o n shows t h a t
Lemma 4 . 3 .
By Lemma 4 . 2 , POe
w.
So the l a s t l i n e i s majorized by
: qn(z)
9 E Aut D.
proof the
is
given
same.
ball.
So t h e r e i s a
< 1 + 2~}. Likewise
be established neighborhood
the
Let
if as in
it
@(P0 ) ~ C can
described dimension
The proof
begins
for
every
he shown that in
the
two.
C
statement
The case
by analyzing
of
4.3.
in
the
of OQ
Em . Fix
a positive
integer
m;
set
~m = ( ( Z l ' Z 2 )
: I z i l 2 + I z 2 1 2 m < 1},
and
3E m 9 P = (1-61,62)
where unit
51, 62 e E outward
norma]
are small
at
P
and non-zero
and
m > O.
with
For w h i c h values
that
QEm(P-tup)
To answer
this question,
let
Re 5 1 > O.
< I + ~?
of
Let t
Up
be the
is it true
153
Notice t h a t f o r
z E W~ ~ 3Em
bounded from
by
0
t h e e i g e n v a l u e s o f t h e Levi form a t
c • ]6212m-2, c
complex l i n e g e n e r a t e d by so t h a t
Vp
and
w2
P
by t a k i n g
wI
t o be t h e
t h e o r t h o g o n a l complex l i n e .
(1+i0, O + i O ) p o i n t s in t h e d i r e c t i o n
:
are
i s a small p o s i t i v e c o n s t a n t .
I n t r o d u c e new c o o r d i n a t e s c e n t e r e d a t
wI
z
Em ~
Vp.
Orient
Let
Fm
be g i v e n by
,.I,(Wl,W2) = (w1/l(~212m,w2/162I ) , where
Fm
i s d e f i n e d t o be
~(Em).
e i g e n v a l u e s o f t h e Levi form on constant
c'
D i r e c t c a l c u l a t i o n t h e n shows t h a t t h e
~(U~) ~ 3F m a r e bounded from
which does not depend on
To a p p l y 11.3 t o
Fm a t
~(P)
0
by some
6 1, 62 .
one may t a k e
~(p)(vl,v2)
= exp v 1, 9 a =TO
d
=
I/2 1 100
where
C,
C'
D
=
C /21 6.
S
=
Cr
are c o n s t a n t s independent of
2m
61 , 62 .
Then 10.3 s a y s t h a t
QFm(~(P-tvp)) < 1 +
provided t h a t @(P-tvp) e {v : l ~ ( p ) { V ) J > bO}
where
b0
may be t a k e n t o be
b0 = I - C " ~ 4 n / l l n 1 6 2 1 l n ~[ .
Write
154
~(P-tvp)
The definition of
@p
shows t h a t
0 < t'
= P'
- t'Vp,
if
< C"' ~ / [ l n l ~ z [ l n
~1
then QF ( ~ ( P ' - t ' V p , ) )
< 1 + ~.
m
Now
Q
i s i n v a r i a n t under t h e b i h o l o m o r p h i c map
@ so i t
follows t h a t
if 0 < t < c"'l~ll~4n/l~.nle;2[~.n e I
then QE (P-tvp) < 1 + e. m
Now i f then
~
~ CC C2
i s a domain s a t i s y f i n g t h e h y p o t h e s e s o f Theorem I . I
i s pseudoconvex.
B(~,a O) n ~
This i s t r u e because i f
i s pseudoconvex.
sequence o f smooth s t r i c t l y
the sets
~ll(u~(1)),
Exhaust
a0 > 0
B(E,a O) n ~
by an i n c r e a s i n g
pseudoconvex subdomains
~21(U£(2) ) ....
i s small then
U1CC U2 O: . . . .
f o r m an i n c r e a s i n g
exhaustion
of
Then
~
by
smooth s t r i c t l y pseudoconvex domains, provided that the subsequence £(j) of increases rapidly. Thus ~ i s p s e u d o e o n v e x . Then a l l o f t h e a n a l y s i s , t h a t i s t h e l o c a l i z a t i o n and c h a n g e o f s c a l e , w h i c h h a s j u s t b e e n p e r f o r m e d on Em a p p l i e s on fl a t p o i n t s P = ( 1 - 6 2 , ~ 2) • ~Q
near
explicit if
1~
(of course
peaking function
~p
from S e c t i o n 9 must be u s e d i n s t e a d
w h i c h can be d e f i n e d
on t h e e g g ) .
0 < t < Co[gl[~4n/l.gn[gllgn
then O ~ ( P - t v p ) < 1 + ~.
Since
§14.
gn[~2[ ~ C n l ~ l l ,
we a r e d o n e .
C o n c l u d i n g Remarks and S p e c u l a t i o n s
~[
It
of the
follows that
155
The t e c h n i q u e s o f t h i s p a p e r a r e v e r y s p e c i a l i n t h e s e n s e t h a t t h e y require ~ to have a nearly transitive group action in the complex normal direction at neighborhood
the distinguished boundary point (1,0). The p s e u d o c o l l a r e d (Lemma 4 . 3 } p r e v e n t s d e g e n e r a c y o f t h e n o r m a l f a m i l y o f maps i n
the (remaining) tangential directions. W h i l e Lemma 4 . 3 i s t r u e i n considerably greater generality, i t i s o f much l e s s u s e i n t h e a b s e n c e o f control over the normal directions. I t w o u l d be v e r y a t t r a c t i v e to have a version of our theorem in which t h e model d o m a i n i s the action
of
any domain of finite
Ant fi
on
fi
extends
type.
In the case of finite
t o an a c t i o n
of
A ut ~
on
~fi.
type, And
restrictions on t h i s a c t i o n c o u l d i n p r i n c i p l e be c a l c u l a t e d from considerations of boundary invariants of CR maps. Of c o u r s e t h e C h e r n Moser-Tanaka invarlant theory has been developed for strongly pseudoconvex domains only; a detailed theory for weakly pseudoconvex domains, even those
of finite type, does not yet exist. A by product of the successful completion of the program indicated in the preceding paragraph might be an understanding of domains with comp@ct automorphism carry
group.
an i n t e r i o r
That is, if
point
P0
~
does not possess enough automorphisms
arbitrarily
near a boundary point
QO'
a nd i f
the obstruction I s l o c a l , t h e n i t s h o u l d be c h a r a c t e r i z e d by a s p e c i f i c finite order jet of the defining function. The n o r m a l f a m i l y t e c h n i q u e s
of
t h e p r e s e n t p a p e r s h e d no l i g h t on t h i s p r o b l e m . Finally, R. Remmert h a s s u g g e s t e d t h a t , i n t h e c a s e o f n o n - c o m p a c t Aut ~ , we a t t e m p t t o r e l a t e t h e r a n k o f t h e L e v i form a t t h e l i m i t p o i n t to the dimension of
Aut ~ .
This will
be a s u b j e c t
for
future
to
Q0
work.
References 1.
F. B e a t r o u s and R.M. R a n g e , On h o l o m o r p h i c a p p r o x i m a t i o n i n w e a k l y p s e u d o c o n v e x d o m a i n s , P a c . J o u r . Math. 8 9 ( 1 9 8 0 ) , 2 4 9 - 2 5 5 .
2.
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INTERPOLATION THEORY IN cn : A SURVEY
Ri ta Saerens Department of Mathematics Michigan State University East Lansing, MI 48824- 1027
O. I N T R O D U C T I O N
This paper is a survey of some known results and open problems in the interpolation theory for various function algebras. First, we introduce some notations and definitions.
We w i l l always use D to denote a bounded domain in ~n.
By Ak(D) (O
denote the algebra O(D) rl Ck(D) of functions holomorphic on D and of class
C k up to
the boundary. A closed subset K of the boundary bD is called an interpolation set for Ak(D) if for each function f in Ck(K), there exists an F in Ak(D) with F = f on K.
It
is called a peak interpolation set for Ak(D) if we can moreover choose an interpolating function F in Ak(D) with the property that IF(x)l < sup K Ill = IIflIK for all x in D \ K , whenever
f
is not identically equal to zero.
constant function
A closed set
K in
bD
on which the
1 can be peak interpolated by a function in Ak(D), is called a peak
set for Ak(D). A closely related concept we will occasionally consider is that of zero set: a compact subset K of bD is a zero set for Ak(D) if there exists an F e Ak(D) such that K = {z e D: F(z) = O}. tt is often useful to consider local versions of the above concepts.
The closed
subset K of bD is called a local peak (resp. local interpolation, local peak interpola-
159
tion) set for Ak(D) if each point in K has an open neighborhood U in E n such that K N 0 is a peak (resp. interpolation, peak interpolation) set for Ak(D).
These sets have been extensively studied for the unit disc in E. The Fatou-RudinCarleson theorem (see 5tout [52], p. 204) states that the classes of zero, interpolation, peak and peak interpolation sets for A(U) coincide and are precisely the subsets of Lebesgue measure zero of bU. For Ak(u) with k ~ I , the situation is quite different. For example, Taylor and Williams [54] showed that peak sets for Ak(u) necessarily finite and that all finite subsets of
(k ~ I) are
bU are peak interpolation sets for
A~(U). On the other hand, Carleson [8] characterized the zero sets of Ak(U) (k ~ 1) as the closed subsets E of bU of Lebesgue measure zero which have the property that bU\E = U Ij with T. Iljl log Ilj1-1 < ~ , where Iljl denotes the length of the open interval Ij.
Some of the results in [:n are similar to those mentioned above for
C but in
general much less is known and we will mainly concentrate on the following questions:
(1) Characterize as much as possible the peak (resp. interpolation, peak interpolation) sets for Ak(D)
(2) When are local peak (resp. local interpolation, local peak interpolation) sets for Ak(D) global peak
(resp. interpolation, peak interpolation) sets for
Ak(D) ?
(3) When are peak (resp. interpolation) sets for Ak(D) necessarily peak interpolation sets or interpolation (resp. peak) sets for Ak(D) ?
(4) When is it true that every compact subset of a peak (resp. peak interpolation) set for Ak(D) is also a peak (resp peak interpolation) set for Ak(D) ?
The techniques used in answering the above questions for k = 0 or k ~=0 are very different.
In the case k = 0 we have Banach algebra techniques and results available.
160
For k = O, ;~ -techniques play a fundamental role. This leads us to restrict our attention mostly to the polydisc Un , strictly pseudoconvex domains and (weakly) pseudoconvex domains and to the cases k = 0 or ~ .
Moreover, for the polydisc we consider only
subsets or the distinguished boundary. (Globevnik in [27] studied more general peak sets for A(Un) .)
Most of the above questions have been fairly satisfyingly answered for the polydisc and for strictly pseudoconvex domains. There are, for example, roughly speaking two types of conditions known that yield characterization results: covering type ones and geometric ones. These conditions are not equivalent but one dimensional manifolds satisfying the geometric ones often satisfy the covering type ones. The covering conditions are usually sufficient ones, while the geometric ones are often necessary and sufficient ones. Most of the above questions are far from having a satisfying answer for pseudoconvex domains.
This paper does not contain complete proofs of the results mentioned. They are all available in the literature.
Sketches of some proofs are given to illustrate the
techniques used.
I. THE ALGEBRA A(D)
I.I
Characterization o_~fpeak, interpolatlon and peak interpolation sets
For the polydisc Un , it is well-known that for closed subsets of the distinguished boundary Tn
the classes of interpolation, peak, peak interpolation and zero sets for
A(Un) coincide (see Rudin [45], p. 132). The same result holds for strictly pseudocon-
161
vex domains. The key step consists in showing that a zero set is also a peak set. This was done by Valskii
[56] for starshaped strictly pseudoconvex domains with
C3-
boundary and then for arbitrary (not necessarily simply connected) strictly pseudoconvex domains with C2-boundary by Ehotlet [15] and Weinstock [59] 1.3).
(see also section
Hence for the unit polydisc and strictly pseudoconvex domains we only need to
study the peak interpolation sets.
This is no longer true for weakly pseudoconvex
domains as we will see later. One of the most powerful tools used in the characterization of peak interpolation sets for A(D) on any domain D is Bishop's theorem (see [4]) which states that if K is a compact subset of bD on which the total variation of every annihilating measure of A(D) is zero then K is a peak interpolation set for A(D).
Davie and Oksendal [18] used Bishop's theorem to prove the following result for strictly pseudoconvex domains
Theorem
: Let D be a domain with such that
bD
C2 -boundary and K a compact subset o f
is s t r i c t l y pseudoconvex in a neighborhood o f
that for each c > 0 ,
K
can be covered by a sequence
bD
K.
Assume
{B(xi,ri)}
of non-
isotropic balls in bD with ~- r i < ~. Then K is a peak interpolation set for
A(D).
(The no.n-isotropic ball B(x,r) = {z ~ bD : I] < r , Iz-xl 2 < r } , where v(x) is the outward unit normal to bD at x .)
The ideas used in proving this result are central to many other results in this area and we outline them briefly.
By combining the dominated convergence theorem and
Bishop's theorem the proof of thls result ls reduced to constructing a uniformly bounded family I1 -
{Fc}c>0 of functions in A(D) such that
lime_~0 Fe(z) = 0 for
z ~ D\K
and
Fc(z)I < c-"~ for z ~ K. For D = {z ~ E n : D(z) < O}, the functions F¢ are obtained as
follows: for c > 0 , let {B(xi,ri)} be the finite cover of K by non-isotropic balls with T_ r i < ~ and let Fc(z) = I - ]'[Gi(z)(<xiri~ - ~ + Gi(z) )-I
where the Gi ~ A(D) are such
162 that Gi(x l) = O, Re Gl(z) ~ milz - xiI 2 on D, Re GI(z) ~,Mllz - xll 2 on bD, grad (Re Gi)(x i) = - grad [~(xi) . By choosing the ci
carefully
(depending on the positive constants m i
and Mi ), the functions F¢ will satisfy all of the above properties. The construction of such strong support functions Gi is rather technical for general strictly pseudoconvex domains (see [18], lemma I for details); for strictly convex domains however, one can easily check that Gi(z) = <xi - z,N(xi)> where N(x I) = < a~p(x i) . . . . . .
a ~ ( x i) >, will do.
Davie and Oksendal also showed that any compact subset of a complex tangential curve in the boundary of a strictly pseudoconvex domain has the above covering property. A manifold is called complex tangential if its tangent space at every point is contained in the maximal complex tangent space of bD at that point. Rudin [46] generalized their result to higher dimensional
(immersed) submani-
folds. (For earlier results in that direction, see Henkin and Tumanov
[31], Nagel [38],
[39].)
Theorem : Let D
be a strictlypseudoconvex domain with C 2 -boundary, ~
an open
set of Bm and ~ :• ~ ~
a nonsingular (:i -mapping with <@'(x)v,N(~(x))>
= 0 for all x ~ ~, v ~ Bm.
Then for every compact subset K of Q , @(K) is
apeak interpolationset for A(D).
(Note that if @(C~) is a submanifold of bD, the stated orthogonality condition is equivalent to saying that @(C2) is complex tangential.)
The proof of Rudln's theorem resembles that of Davie-Oksendal's result. However, the construction of the corresponding F~ is less straightforward even in the strictly convex case. We refer the interested reader to [46].
Nagel and Rudin [40] (see also Nagel [39] for the unit ball) studied the intersection of a peak set for A(D) for any El-domain D with a transverse curve; ie., a curve in bD whose tangent space at every point is not contained in the maximal complex tangent space of bD at that point. They showed that if V Is a transverse curve of class (:2 in bD, then any bounded holomorphic function on D has nontangential boundary
163
values [py]-almost everywhere, where py
is the measure on bD defined by
If dp =
If(~(t))dt for all f e C(bD) and any parametrization ~ of y. However, it is easy to check that if G ~ A(D) is a peak function for some set K, then F = exp[ilog(l-G)] is a bounded holomorphic function on D with no limit along any curve in D that ends at a point of K. Hence, a peak interpolation set of a strictly pseudoconvex domain intersects any transverse curve y
in a set of [jJy]-measure zero. Combinig this with Rudin's
result, we have a complete characterization of those peak interpolation sets for A(D) which are contained in a submanifold of the boundary of a strictly pseudoconvex domain. Stout [53] proved that the topological dimension of any peak interpolation set for A(D) for a strictly pseudoconvex domain in C n is at most n-I and one might ask if a converse of Rudin's result holds, namely is every peak interpolation set for
A(D)
(locally) contained in (n-i)-dimensional complex tangential submanifolds of bD ? The answer is no: Henriksen [32] showed that the boundary of any smooth domain D in En contains a peak set for A(D) of Hausdorff dimension 2n-l. (Tumanov [55] constructed one of Hausdorff dimension 2.5 in the unit sphere of £3 .) This example indicates that the question of completely characterizing all peak interpolation sets for A(D)
for a
strictly pseudoconvex domain is far from trivial.
The covering condition for the polydisc
Un similar to Davie-Oksendal's one for
s t r i c t l y pseudoconvex domains is the null S-width condition due to Forelli E has null S-width
[20].
A set
(where S is a set of unit vectors in tRn) if for all ~ > O, there
exists a family [uj} in S anda family [Ij} of open intervals in IR such that ~- IIjl < and E c O { x ~ l R n ' < x , u j > ~ l ] } .
(Recall IIjl denotes the length of Ij.)
Theorem : Let G be a countable union of sets of null S -width where
S
varies over
compact subsets of unit vectors in Fin+. Then, for every compact K of G,
exp(K) = [(eiXl,..-,eixn): x ~ K} is apeak interpolationset for A(un).
By Bishop's theorem, it is sufficient to show that
Ivl(exp(K)) = 0
for all compact
subsets K of G and all measures v annihilating A(un). It is easier to l i f t the problem to Fin that to work on T n . The Riesz representation theorem yields for any complex
164
measure v on T n a measure X on ITtn by requiring that for all compactly supported functions g e d(IFIn), Ig dX = I~- g(x)[Tl4/xjsin(xj/4)] 2 dr(y) where the sum is taken over all x = (Xl . . . . . Xn) e exp-l(y).
It is easy to check that a measure v annihilating
l i f t s to a measure ~ such that Jexp(i<x,y>) d,~(y) = 0 for all
A(U n)
x ~ I~n+ and that hence, it
suffices to prove that Ikl(K) = 0 for all compact subsets K of G and all complex Borel measures ~ on I~n with the above property.
This follows from showing that for all
functions g in a suitable dense subset of L2(I>,I) the estimate Ii u Igl2 d>, ~ C(5,g)Ilul IIglfL2(Ixl) holds for some constant C depending on S and g and for t u= {x~ ~n: <x,u> ~ l} and u ~ S (seeRudin [45],pp. 140-147 orForelli
[]8]).
Geometric results similar to those of Rudin also exist (see Saerens [48]). The submanifolds of the distinguished boundary -Irn that occur are those which satisfy one of the cone conditions introduced by Burns and Stout [71: a submanifold M of
l r n satls-
lies the closed cone conditioq if at every point of M the tangent space to M does not intersect the open positive cone in the tangent space of
"IT"n at that point.
It satisfies
the open cone condition if at every point of M the tangent space of M intersects the closed positive cone in the tangent space of
Tn
trivially.
As was the case for the
geometric and covering type conditions for s t r i c t l y pseudoconvex domains, curves in "lrn satisfying the open cone condition also satisfy the null S-width condition of Forelli's result. This need not be true for higher dimensional submanifolds
(see 5aerens
[48]).
The analogue to Rudin's result is as follows:
Theorem
:
Let M be a C 2 -subman/foldof qrn wh/ch satzsfies the open cone condit/on. Then every compact subset of M Conversely, if M
/s a peak interpolation set for A(un).
is such that every compact subset of it/s apeak interpo-
lation set for A(U n) then M satisfies the closedcone condition
(As was the case w i t h Rudin's result, this theorem holds for immersed manifolds.)
The key remark in the proof of the f i r s t part is to note that the manifold M can be mapped, locally at any point p of M, into a complex tangential manifold of class
¢ I in
165 the boundary of the unit ball of E n . Using this mapping, Rudin's proof for the similar result on the ball can be transcribed to the polydisk. The proof of the second part is similar to the NageI-Rudin result mentioned before: A manifold
M
which does not
satisfy the closed cone condition has the property that for every point p of M, there is a nontangential curve ¥p : [0,1) -, Un ( ie, a curve whose projections in each coordinate plane are nontangential) with limt_,l ~/p(t) = p such that for every bounded holomorphic function F on Un , limt_~l F(yp(t)) exists for almost all p in M (with respect to the natural measure on M ). Hence by the remark made previously, compact subsets of such an M cannot be peak interpolation sets for A(D).
Problem
: Find necessary and sufficient conditions for a (smooth) submanifold of
T
to have the property that all its compact subsets are peak interpolation sets for A(Un).
For pseudoconvex domains much less is known about the characterization of peak, interpolation and peak interpolation sets for A(D). It is not true (as it was for strictly pseudoconvex domains and for the polydisc) that the classes of these sets coincide (see also section 3). Since for any domain D with C2-boundary the peak points for A(D) are contained in the closure of the set of strictly pseudoconvex boundary points
(see
Basener [2]), not every interpolatlon set for A(D) is a peak set for A(D) even if the domain is pseudoconvex. The question of when every boundary point of a pseudoconvex domain is a peak point for A(D) is far from completely answered. Bedford and Fornaess [:3] proved the following
Theorem
: Everyboundarypoint of a domain D z;o E2 is apeakpoint for A(D)
if D is
(a) a pseudoconvex domain with real analyticboundary or (b) a pseudoconvex domain of finite type with C ~ -boundary.
If D is apseudoconvex domain with real analyticboundary in [n then boundarypoints where the Levi form has corank I are peak points
166 A domain D in C2 is of finite type if there is a finite bound on the order of contact one-dimensional complex manifolds have with
bD.
In £n, there are several distinct
concepts of "finite type" (see for example Bloom [5] and d'Angelo [17]).
Problem
: Let
D be a smoothly bounded pseudoconvex domain in
supremum of the order of contact one-dimensional
En for which the
complex varieties
(or
complex hypersurfaces) have with bD is finite. Is then every boundary point of D a peak point for A(D) ?
Bedford and Fornaess
[3]
proved moreover that in the case of a pseudoconvex
domain in E2 with real analytic boundary, the peak functions can be chosen to vary continuously with the peak point. This was generalized by Fornaess and Krantz [23] to any compact metric space X and closed subalgebra of C(X).
Because of the above remarks it is clear that complex tangential manifolds cannot play the same role in the interpolation theory for A(D) of pseudoconvex domains as they did for strictly pseudoconvex domains. However, using Whitney covers, del Castillo [9] proved a result very similar to the covering result of Davie and Oksendal
[18] for
smoothly bounded convex domains containing no line segment in their boundaries. A Whitney cover for a subset E of bD is a family {B(xi,rj)} of non-isotropic balls which are pairwise disjoint and for which there exists constants k and h such that E is contained in the union of the
B(xj,krj) , every point of
E
belongs to at most
differrent balls B(xj,kr]) and every B(xj,hrj) intersects bD\E.
M
(Any open set in the
boundary of a domain D with C2-boundary has a Whitney cover, see Coifman and Weiss [I6], p.70.)
T h e o r e m : Let D be a convex domain with a C 2 -boundarynot containing line segments Let K beaclosedsubsetof bD
with theproperties that K haszeroarea
measure and bD\K has a Whitney cover {B(xi,ri)}with ~ rj < oo. Then K is a zero set andapeak set for A(D).
167 The proof bears a strong resemblance to that of a theorem of Chollet in [14] which states that a similar metric condition on a Whitney cover for bD\K when D is s t r i c t l y pseudoconvex, implies that K is a zero set for A~(D) (see section 2.1). The properties on D and the fact that Re <xi-z,v(xi)> > 0 on Dk{xi} imply that the function <~(Z) = ~- ~kiri(ri+<xi-z,v(xi)>) - 1 where ,ki are positive numbers tending to infinity and such that ]E ,kir i < ~ , is holomorphic on D, continuous on D\K and tends to infinity whenever z ~ bD\K tends to a point of K. The function exp (-~(z)) extends to a function F ~ A(D) which vanishes exactly on K. Since convex domains are simply connected, zero sets for A(D) are clearly also peak sets.
Both Davie-Oksendal's and del Castillo's constructions work on any domain
D
for
which there exists a function G defined on bD x D such that G(x, • ) ~ A(D) for all x in bD,
G(x,x) = 0 , grad (Re G(x,x)) = - grad g(x) , Re G(x,z) is estimated from below on
bD x D
by
mI¢-zl 2 or mlI for some positive m
and from above on bD x bD by
Ml~-zl 2 or Mll for some positive M
P r o b l e m : For which domains does such a function H exist ?
del Castlllo also shows that any compact subset of a complex tangential curve of class (:2 satisfies the above conditions, which leads to
P r o b l e m : Is it true that every closed subset of a complex tangential submanifold of bD
is a peak set for
A(D) when D is a smoothly bounded convex domain not
containing line segments in its boundary?
del Castillo's result illustrates the importance of a suitable generalization of the concept "non-isotroptc ball" to more general pseudoconvex domains.
168
1.2. Local versus global
Since finite unions of peak sets for A(D) are clearly peak sets, local peak sets are always global peak sets.
Problem : Let
D be a pseudoconvex domain, are local interpolation sets for
AiD)
necessarily global ones ? (Same question for local peak interpolation sets.)
1.3.
Implications
It is clear from the definitions that any peak interpolation set for A(D) is also an interpolation, peak and zero set. What other such implications hold for pseudoconvex domains (recall the remarks made in 1.1 ) ?
We have the following result clue to Varopoulos
[57] which emphasizes the
importance of studying when every boundary point is a peak point for AiD).
Theorem : Let D be anyboundeddomain in £n and K a compact interpolationset for
A(D) of which everypo/nt is apeakpo/nt for A(D). Then K is apeak interpolation set for AID).
Glicksberg [26] gave an easy proof of this theorem along the following lines: since K is an interpolation set, the open mapping theorem guarantees that there exists a positive constant <x such that every f ~ C(K) extends to an F ~ AiD) with IIFIID ~ cdlrllK. Fix now E> O,ameasure
v which annihilates A(D) and an open set Q with K c Q
and Ivli~\K) < c . Since every point of K is a peak point for
AID), one can choose
functions f l , - - . , fN in A(D) such that Ilfjll = 1 , Ifj(z)l < c for z ~ D\Q and that the
169 sets Ej = {z e D : l l - f j ( z ) l < c] cover K. Let Kj be pairwise disjoint compact sets w i t h Kj c Ej 13 K and
Ivl(o\ UKj)<
2~ . For each j , choose functions
gk e A(D) with
ll&ll
<
c( and gk(z) = ~akj on K j , where £a is a fixed Nth-root of unity. Since the functions h m = N- i Z~a-kmg k are one on K m and zero on Kj with is such that IIHII < cc2, I1 - H(z)l < ~ on UKj
j,,m, the function H = Zhm2f m
and IH(z)l < oc2c on D \ ~ .
construction for Sn-* 0 yields a uniformly bounded sequence of functions
Repeating this [Hn] in A(D)
which converges pointwise to the characteristic functions of K. The conclusion of the theorem follows then from Bishop's theorem.
Jimbo [36] studied the question as to which conditions on the set of weakly pseudoconvex domain would imply that a peak set for A(D) is necessarily a peak interpolation set. The proofs he gives make use of a theorem of Fornaess and Nagel [24} which is not correct under the generality statea. Jimbo's proof is correct whenever the theorem of Fornaess and Nagel holds (for example, when the set of weakly pseudoconvex boundary points is totally real or when D is in E;2 with a real analytic boundary). 5ince no counterexample to Jimbo's result is known, we state it as a problem.
P r o b l e m : PROVE OR DISPROVE : Let D be a pseudoconvex domain with C~-boundary in
Cn. Assume that the set o f weakly pseudoconvex boundary points consists o f peak points for A(D) and is a stratified totally real set, Theneverypeak set for A(D) is apeak interpolation set for A(D).
A set is called a stratified totally real set if it is the disjoint union of sets
Mj
where each Mj is a totally real C=-manifold of £n\ U i<j Mi-
Problem
: If the above statement is false, find other conditions on the domain for which
such a result holds.
For simply connected domains a zero set for A(D) ls always a peak set since If f is a function with Ilfll < 1, exp(1 - Iogf(I-logf) - I )
is equal to one on [z: f(z) = O} and
170
strictly less than one elsewhere. Chollet [15] obtained the same result for an arbitrary strictly pseudoconvex domain D with
C2-boundary by constructing at every point
p ~ bD, a simply connected strictly pseudoconvex domain Dp E D such that bDp and bD agree near p. This yields a local peak function (on Dp) which can be extended to D by standard
~-technlques.
These
~-technlques do not hold on general pseudoconvex
domains. Verdera [58] obtained the same result for pseudoconvex domains by using an additive Cousin problem.
1.4. Comoact subsets of peak sets and peak interpolation sets
Compact subsets of interpolation (resp. peak interpolation) sets are clearly again interpolation (resp. peak interpolation) sets. By the remark made in
1.1 the same
holds for peak sets in the case of the unit polydisc or a strictly pseudoconvex domain. It seems to be s t i l l an open question whether this is true for pseudoconvex domains.
2. THE ALGEBRA A==(D)
2.1 Characterization of Peak. interoolation and peak interpolation sets
Compact subsets of complex tangential submanifolds of the boundary of the unit ball B are in general not peak interpolation sets for A~(B) lemma, Noell [43] showed
(see 5aerens [48]). Using Hopf's
171 Theorem
: Let D be a bounded domain with C 2 -boundary end K a peak interpolation set for Ak(D) ( 1
Problem
Then K is finite.
: Characterize which finite subsets are peak interpolation sets for
Ak(D)
when D l s a (strictly) pseudoconvex domain,
There are no results known for differentiable peak interpolation on the polydisc which yields immediately the following questions:
Problem
: Are closed subsets of manifolds in "11 "n which satisfy the open cone condi-
tion (local) peak interpolation sets for Ak(Un) ? Are there any infinite peak interpolation sets for Ak(Un) ?
For strictly pseudoconvex domains, complex tangential submanifolds of the boundary play an important role in the study of peak and interpolation sets for Am(D) just like they did for A(D). We have the following result:
Theorem
: Let
D be a strictly pseudoconvex domain with
complex tanqential subman/fold of class subset K of M
C ~° -boundary and
C ~° of
bD .
M
a
Then every closed
is apeak set andan interpolation set for A~°(D).
Hakim and Sibony [30} proved the interpolation and local peak results. They follow from the following technical lamina: For every p ~ K, there exist neighborhoods U and V of p , with
V compactly contained in U , and a function
following properties : (i) ~ = 0
on
K n v ;
( i i ) Re ~p < 0
on
with the
(D (I U) \ (K (I V) ;
(iii) D ~ ( ~ ) = 0
on K ( I V for all <x; (iv) ~(l/
to C ~ ( D O U ) ;
(v) for every F ~ C~(D(1U) with D~(aF) = 0 for all
<x , (l/~p)aF
extended to K rl V by zero, belongs to C~(D n U). (It is in the proof of this }emma that the condition on the tangent space of M is used.) The local peak result follows easily. Indeed, choose a C~-function × with compact support in U and equal to one on V. Let g be a solution of a u - a ( x / ( p ) . Then h : ~ / ( × - g ~ p )
is zero on K r l V
and Reh <0 on
172
D\(K fl V). The local interpolation result is obtained similarly. The global one follows from this by a partition of unity argument.
The global peak result does not follow so
readily from the local one since finite unions of peak sets for Am(D)
are not always
peak sets (see Hakim and Sibony [30]). Chaumat and CholIet [I I] obtained the global peak result by showing that for any compact subset K of a complex tangential manifold M there exist a neighborhood U of K In IEn , a strong support function ~ ~ C~(U) and a positive constant c such that (I) K ={z~U:~(z) =0}; (il) D ~ ( ~ ) = 0
on U n M
for
all <x ; (iii) Re ~(z) 2 c [dist(z,M)]2 for all z ~ U n D. To obtain a peaking function on K,picka
C~-function ~
with support in U suchthat
0 < ~ < I and ~ = I in a
neighborhood V of K. Then a(wlcp) extends to a closed (0,1)-forrnof class C ~ in D. Let
u
be a (t~-solution of
~u = ~(~vl~) . By adding a positive constant to the
holomorphic function v = W/~ - u, we may assume Re v > 0 on D\K and hence exp(I/v) peaks on K. Chaumat and Chollet [I2] also obtained a partial converse
Theorem : Let D be a strictlypseudoconvex domain with C a -boundary and let K be a closed subset of bD
which is a localpeak set for A2(D).
Then
K
is
locally contained in a complex tangential submanifold of bD.
The above result raises the question whether global peak sets are globally contained in complex tangential manifolds, Chaumat and Chollet [13] showed the answer is negative rot n ~ 4. They construct a compact K of some open set U of [;3 such that K is the zero set of a positive plurlsubharmonlc function in C~(U) and is not contained tn any proper submanifold of U. The compact K is then naturally imbedded in the boundary of a strictly pseudoconvex domain D in E 4 such that contained in complex tangential submanifolds of
K is locally but not globally
bD . Fornaess and Henriksen
[22]
proved that for n = 3 , peak sets for A-(D) are always globally contained in complex tangential submanifolds by first proving the existence of a stratification by complex tangential manifolds whose union contains K. When n = 3, this stratification yields the existence or a complex tangential manlfold containing K.
173
The polydisc case shows the same simililarity between k = ~
and k = 0 as the
strictly pseudoconvex one does (see Saerens [47]).
Tlleorem : Let M be a C ~ -submanifold of the distinguishedboundary of the unitpolydisc U n which satisfies the open cone condition Then every closed subset of M
is a loca/peak set andan interpolationset for A~(Un).
The local interpolation and peak results are obtained with the same methods as in the case of A(Un) : M is locally mapped into a smooth complex tangential manifold in the boundary of some strictly pseudoconvex domain with
Coo-boundary. The global
interpolation result is obtained by a partition of unity argument. This embedding method does not however yield a global peak result.
A complete characterization of peak sets
for A~(U n) is known.
Theorem
: Let K
be a compact subset of the distinguishedboundary T n 0 / the unit
polydisc U n . Then the following are equivalent (a) K
is apeak set for Aoo(un) ;
(b) there is an open neighborhood ~
of K m
T n anda closed
C °O-submanifold M of (Q which satisfiesthe open cone condition andcontains K.
The tact that (a)
implies
(b) was proved by Saerens and 5tout
[49] by using
Hopf's lemma. It was also shown there that it might not be possible to take for M a closed submanifold of the whole l "n . As mentioned earlier a local version of the converse implication was proved by Saerens [48].
Labonde [37]
proved the global peak
result by using ~-techniques on the polydisc similar to those used by Chaumat-Chollet [ I 1] in the case of strictly pseudoconvex domains.
We recall
(see section 1.1) that Bedford and Fornaess
[3]
showed that every
boundary point of a pseudoconvex domain D in £2 with either a Coo-boundary and of finite type or with real analytic boundary is a peak point for
A(D). This is in some
174
sense the best possible result for such domains: Fornaess [21] constructed a pseudoconvex domain w i t h real analytic boundary which has only one weakly pseudoconvex point and such that it is not a peak point for AKD).
The study of the existence of differen-
tiable peak functions requires looking at stronger
"finite type"
conditions than those
defined in section 1.1 : Let 0 be a boundary polnt of a smoothly bounded domain D in Cn w i t h defining function p and assume the tangent space to bD at 0 is given by Re z n = O. Then 0 is a point of
strict finite type
if there are a function
h holomorphic in a
neighborhood V of 0 in l~n-l, a positive integer k and a positive constant m such that h(O) = 0 and @(w,h(w)) ~ m Iwl k on V . boundary point p
[29]
showed that a
of strict finite type is a local peak point for At(D) (Le., there is a
neighborhood U of p in [5]
Hakim and Sibony
such that some function F in At(D ilU) peaks at p ).
Bloom
showed that this s t r i c t finite type condition is not sufficient to yield an
A~(D)
peaking result. (He also Introduced a stronger type condition necessary and sufficient for a boundary point of a pseudoconvex domain to be a local peak point for functions holomorphic in a neighborhood of the point.)
P r o b l e m : Are there type conditions that are necessary and sufficient for a point to be a
(local) peak point for A~(D) ?
Noell
[41]
showed that for t > 0 sufficiently
D = {(z,w) ~ C2 : Rew + Izl 4 - t(Imz) 4 + (Imw) 2 < O}
small,
near
0
the convex domain has the property that
M = {(z,w) ~ bD : Imz = Imw = O} is a complex tangential curve such that M n U is not a peak set for A-(D)
for any neighborhood U of 0 .
He also pointed out that for
D :
{(z,w) ~ C2 : Izl 4 + Iwl 2 < 1} the set K I U K2 where K 1 = {(z,w) ~ bD: Imz = Imw : O} and K2 = {(z,w)~ b D : R e z = Imw = O} is a peak set for contained in any complex tangential submanifold of
A-(D) bD .
which is not
(locally)
Hence complex tangential
manifolds do not play here the same role as they do for s t r i c t l y pseudoconvex domains. Complex tangential conditions at certain points however s t i l l do play a role. Iordan [33] obtained some necessary conditions for local peak sets for A'~(D) involving them.
Theorem:
Let D be adomain with C2-boundary, K a localpeak set for A2(D) and p a point in K
where bD
is pseudoconvex and where the Levi form has q
175
zero eigenvalues. Then there is aneighborhood V
of p andan (n+q-l)-di-
mensional C i -submanifold M of V flbD such that (a) K fl V is contained in M ; (b) M
is complex tangent/a/at aNpoints x ~ K ;
(c) the complex dimension of the maximal complex tangent space of
M at anypoint is at most q.
The f i r s t
example of Noell mentioned above also shows that the necessary
conditions in Iordan's theorem are not sufficient.
Iordan proved an analogue of Hakim and
Sibony's technical lemma mentioned before in this section and used it obtain sufficient conditions for a set to be a local peak set for Am(D) in terms of s t r i c t l y q-convexity and the existence of
£R-functions with certain growth estimates
(see
[33]
for
details).
2.2 Local versus global
Fornaess and Henriksen
[22] showed that for a strictly pseudoconvex domain
D
with C~-boundary local peak sets for A°°(D) are always global peak sets by proving the following technical lemma: Let M 1 , M 2 be complex tangential submanifolds of bD with d i m M I < d i m M 2 and M I rim 2 open in M i . Then any compact subset K of M i U M 2 such that K f l M i is open in K, isapeakset for A~°(D). The peak function for K
is
obtained by "gluing" the peak functions for K ~I M i and K fl M 2 . Using this lemma repeatedly yields that local peak sets are global ones.
This argument cannot be generalized to arbitrary pseudoconvex domain. shown by Noell [41] • the domain
This was
D = {(z,w) ~ £2 • p(z,w) = Iw+eilnz~12 -1 + C(Inzz) 4}
(where C is a large positive constant)
is a pseudoconvex domain such that its set of
weakly pseudoconvex boundary points {(z,w) ~ bD :[zl - I , w = O} is a local peak set for Am(D) but not a global one. The reason the set fails to be a peak set is because there are
176 real analytic complex tangential manifolds contained in the set of weakly pseudoconvex boundary points. A pseudoconvex domain D in £n with
C°°-boundary has the (NP)-
property if the set of weakly pseudoconvex points of bD is contained in finitely many real analytlc curves and does not contain any real analytic, complex
tangentlal
manifolds. For example, any convex domain with real analytic boundary in Cn has the (NP)-property if n = 2 (see Noell [41]); this is false for n ~ 3.
Theorem : Let D be apseudoconvex domain with real analytic boundary in C 2 which
has the (NP) -property Then any Iocalpeak set for A°°(D)
is a peak set for
A~(D).
Noell [4t] proved this by patching local peak funcions. This patching cannot always be done at weakly pseudoconvex points but the hypotheses on the domain guarantee it need only be done at strict pseudoconvex points.
Problem : Find conditions for n > 3 which imply the same result.
The question of whether local peak sets for A~(U n) are necessarily global ones, is still open.
As mentioned in section
2.1, Hakim and 5ibony
[30]
used a partition of unity
argument to obtain global interpolating functions from local ones in the case of smoothly bounded strictly pseudoconvex domains (see Saerens [48] for the polydisc case). This argument works for any domain D and compact subset
K of
bD which is a local
interpolation set and has the property that every compact subset of K is a peak set for Am(D) .
177
2.3 Implications
For the function algebras A°°(D) of a smoothly bounded domain, peak sets or interpolation sets are rarely peak interpolation sets since as we saw in section I the peak interpolation sets for Ak(D) (for l;k~oo) are necessarily finite sets (see Noell [43]), while peak or interpolation sets need not be finite.
Problem: ls there any general result of the following form possible for the polydisc: If K is a peak set and an interpolation set for A~°(Un), then it is a peak interpolation set for A°°(Un) ?
There remain the questions whether being a peak set for
Am(D) implies being an
interpolation set for it and conversely, whether being an interpolation set for
A°~(D)
implies being a peak set. For strictly pseudoconvex domains, the answer is affirmative to both questions because of the characterization of these sets given before.
For the
unit polydisc, it is known that if the subset K of the distinguished boundary is a local peak set for A°°(un) then it is an interpolation set for A°°(U n) (see Saerens and Stout [49]). If the domain D is pseudoconvex only partial results are known. Interpolation sets for Am(D) need not be peak sets even for A(D) since not all boundary points are peak points. 5ome conditions are known under which being a peak set implies being an interpolation set. They are due to Noell [42].
Theorem: Let D beapseudoconvexdomam m
~2
with C~°-boundaryandof f/nite
type and let M be a C ~° -curve m the boundary which is a localpeak set for
A~(D). (a) /f bD is of constant type along M, then every closedsubset of M is an interpolation set for A~°(D).
(b) If bD and M are real analyt/c, then every closed subset of M is an interpolation set for A°°(D).
178 The proof of this theorem is similar to the proof of the interpolation result of Hakim and Sibony for strictly pseudoconvex domains discussed in section 2.1. This sort of argument requires the existence of a strong support function
~
for which
Re~
vanishes only to finite order along M. 5uch functions always exist when the type of bD along M is constant. If the order is not constant this need not be the case. Indeed, for D = {(z,w) ~ C2 : l z P + Iwl 2 < 1} the set M = [(z,w) ~ bD: Imz = Imw = O} is a peak set for A~(D) but any strong support function vanishes to infinite order at the points (0,+ 1) in the direction of [(z,w) ~ bD: Rez = Imw = O}. These points (0,_+1) are of type 4 while all the other points of M are of type 2.
The above argument can be modified for points
near which the type is not constant under the additional hypothesis that both bD and M are real analytic.
Noell also gives an example of a convex domain with
C~-boundary whose set of
weakly pseudoconvex boundary points is a line segment which is a peak set for
A'~(D)
but is not a local interpolation set for A°~(D). This set consists of points of infinite type.
Problem
:
Find similar results for [n with n ) 3.
Recall
(see section 2.1)
that for strictly pseudoconvex domains
(resp the unit
polydisc) a compact subset of the boundary (resp the distinguished boundary) is a local peak set for
A~(D)
if and only if it is locally contained in a complex tangential
submanifold (resp. a submanifold satisfying the open cone condition).
This is not true
for pseudoconvex domain and Noell's result indicates that being a local peak set is the key factor in obtaining interpolation results (see also Saerens and Stout [49] polydisc case).
for the
179
2.4 Compact subsets of peak sets and interpolation sets
Compact subsets of interpolation sets are clearly again interpolation sets.
The question whether closed subsets of peak sets are again peak sets, is in general less trivial.
By the characterizations given of such sets in section 2.t
the answer is
clearly affirmative for strictly pseudocovex domains and for the unit polydisc. pseudoconvex domains, this is not always the case. Noell [44] of a convex domain D with
For
constructed an example
(~-boundary for which the set K of weakly pseudoconvex
boundary points is a peak set for A~(D) but it contains a point that is not a peak point for any Ak(D), (k>O). His example is such that the set K is actually a line segment. Positive results due to Noell and to Iordan are known under various additional conditions on the domain to guarantee the above phenomenon cannot happen.
Theorem: Let D be apseudoconvexdomain. Then D has thepropertythat any closed subset L of apeak set K for A°°(D) is also apeak set for Am(D)
if D
satisfiesany of the followmg properties :
(a) D is contained in C2 andhas area/ analyticboundary (see Noell [41]); (b) D is smoothlyboundedandof finite type m (c) D has a ~
£ 2 (see Noell [4i]);
-boundary and the (NP)-property (see Iordan [35]).
For the proof of this theorem one first notices that the methods used by Chaumat and Chollet in [12] for strictly pseudoconvex domains can be used here to show that the set L I = (K I~ w(bD)) U L is a peak set, where w(bD) is the set of weakly pseudoconvex points of
D . To prove
(c)
Iordan shows first that if
p
is an isolated point in
(K ~1 w(bD))\L, L I \ { p} is also a peak set. This can also be used to eliminate points of K (! w(bD) near which w(bD) is contained in real analytic curves which are not complex tangential at p. Indeed, such points have a neighborhood U such that K ~ w(bD) (I U is finite. Hence only non-isolated points p of K fl w(bD) (I ¥ for a real analytic curve y
180 complex tangential at p remain. The (NP)-property ensures that p is the limit (from both sides) of a sequence of points of
y
at which
y
is not complex tangential.
Deleting appropriate neighborhoods of such points using the previous step yields a peak set L2 such that p is an isolated point of L2 n y . If p belongs to another such curve, the procedure is repeated; otherwise p can be excised from L2 by the previous step. Noell proved (a) by combining similar "cutting out" ideas with the fact that w(bD) can be decomposed in a union of pairwise disjoint real analytic manifolds SO , S 1 and S2 with the properties that
(i)
each Sj consists of finitely many j-dimensional real
analytic totally real manifolds;
(ii)
S 1 is closed in bD\S 0
and 52
is closed in
bD \ (50 U S1); (iii) each component of S2 consists of points of the same finite type. This decomposition result due to Fornaess and Ovrelid [25] is only available for domains with real analytic boundaries. Noell used another decomposition theorem due to Catlin [ 1O] for domains of finite type.
Problem : Find other sufficient conditions on D for the above property to hold (especially for domains in ~n with n~3).
3. MISCELLANEOUS
For the sake of completeness
we mention without
interesting interpolation questions and results.
any details some other
181
3.1 The algebras Ak(1)) with 0 < k <
When interpolating Ck-functions one has to expect to lose some degree of smoothness (see Rudln [47] and Steln [50]). Therefore we introduce the following concept: a closed subset K of the boundary of a domain D is an (s,k)-interpolation peak interpolation)
set if every
dS-function
(different from zero)
(resp. (s,k)-
on
K
can be
interpolated (resp. peak interpolated) by functions in Ak(D).
Noell's result state that for any bounded domain with
C2-boundary, the (k,k)-peak
interpolation sets (for l~k<~) are necessarily finite sets (see Noell [43]).
Problem: Characterize the (s,k)-peak interpolation sets.
For the polydisc we have (see Saerens [48] closed subsets of a (t(~-submanifold of
"]rn
and Saerens and Stout
[49])
that
satisfying the open cone condition are
(s,s- I )-interpolation sets.
The methods used by Hakim and Slbony [,30] for k = oo yield that compact subsets of smooth complex tangential manifolds are (s,k)-interpolation sets, where k = [s/2] - I if s is odd and k = [ s / 2 ] - 2
if s is even. It is known that a loss of the order of half
the derivatives is unavoidable in this situation
(see Rudin [47]
and Saerens [48]).
In
the case of a pseudoconvex domain the loss of differentiability is even bigger.
P r o b l e m : Describe the minimal loss of differentiability the domain.
in terms of the geometry of
182 The example that Noell [41] gave to show that local peak sets for A~(D) need not be global peak sets can be strengthen to show that local peak sets for A~(D) need not even be peak sets for A2(D).
Problem : Are local peak sets for Ak(D) global peak sets if the domain has the (NP)property ? Are there weaker conditions on the domain which yield this ?
Related to this is the interpolation of functions which are Ck and which derivatives of order k satisfy Lipschitz conditions of order m. This was studied by Saerens and Stout in [49] for the polydisc and by Bruna and Ortega in [6] for the ball.
3.2 Norm preserving interpolation
Another interpolation concept which might be useful to study is norm preserving interpolation. A closed set K in the boundary of a domain D is called a norm preserving interpolation set for Ak(D) if for every function f in Ck(K) there exists a function F in Ak(D) which equals f on K and with IF(z)l < sup K Ifl for all z ~ D\K. This has been studied for k = 0 by Globevnik [28] in the case of the polydisc.
Problem: Study the norm preserving interpolation sets for k > 0 for the polydisc. What can be said about norm preserving interpolation sets for pseudoconvex domains ?
(strictly)
183
3.3 M a x i m u m modulus sets
We call a closed subset K In the boundary of a domain D a maximum modulus set for
Ak(D) i f there exists a function f in
Ak(D) such that Ifl = 1 on K and Ifl < I on
D\K. (The concept of local maximum modulus set is defined similarly.)
Duchamp and
Stout [19] showed that i f D is a s t r i c t l y pseudoconvex domain in E:n and M is an ndimensional ¢2-submanifold of bD which is a maximum modulus set for A2(D) then M is t o t a l l y real and foliated by compact complex tangential manifolds. They also have a partial converse. Iordan [34] showed that a closed subset of the boundary of a s t r i c t l y pseudoconvex domain D which is a local maximum modulus set for
A~(D)
is locally
contained in a t o t a l l y real manifold which is foliated by complex tangential manifolds.
Problem : What can be said about (local) maximum modulus sets for pseudoconvex domains and for the polydisc ?
3.4 Interpolation of order s
A natural concept in the interpolation theory of C~-functions is that of interpolation of orders ( O ~ s ~ ) .
Aclosedsubset K of bD is an interpolation set of o r d e r s
i f for each function f in £~(K) with
af = 0 vanishing to order s-1 on K there is a
function F in A~(D) such D~F = D~f on K for all multi indices c( with 0 ~ I~l < s . Chaumat and Chollet [12] showed that for a s t r i c t l y pseudoconvex domain a peak set for A~(D) is an interpolation set of order s for all 0 < s < ~ convex domains mentioned in section 2.3
Noell's result for pseudo-
also gives interpolation of infinite order.
184 4.
[I]
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EXTENDABILITY OF HOLOMORPHIC FUNCTIONS Berlt Stensones* Rutgers University Mathematics Department New Brunswick, New Jersey
Introduction In this paper we are studying the one side of a real hypersurface in
extendability of holomorphic functions from
C n.
The proofs presented here is highly based on the maximum principle and the subaveraging property of plurisubharmonic Let
S: = {p g C n : r(p) = 0}
two sides
S+
and
S-
of
S- := {p E C n : r(p) < 0}. U+: = S+ ~ U If in
U
and
S
where
functions. r
is a C 2
we shall mean that
If
U
defining function.
S+: = {p e C n : r(p) > O}
is a neighborhood of a point
and
p g S, then
U-: = U ~ S-.
is small,
then this makes sense, i.e.
U-
and
U+
are open domains
C nWe shall be interested in finding out whether holomorphic
U-
By the
has holomorphic extension over
functions on
U+
or
p.
Our main result is a generalization of a result by J.M. Trepreau
[1] saying:
THEOREM i [i]: Let or
Poe
S
be a
U-
C2-smooth hypersurface in
for every neighborhood
complex hypersurface in By
~
S
U
through
Cn
of Po
and let
Poe
S.
Then
Poe
U+
if and only if there is no germ of a
P0"
we mean the envelope of holomorphy of a domain
D.
In fact Trepreau is able to prove that if there is no germ of a complex hypersurface in Let C n.
V
S
through
Po
then the following holds:
be a neighborhood of
Po in S
Then there exists a neighborhood
W
and let of
Po
*The author partly supported by an N°S.F. grant.
in
U
be a neighborhood of C n, W
is independant of
V
in U
I90
(i.e.
W
is independent of how thick
U
is) such that
W ~ ~+
Note that we do not know whether
or
W C ~
W~,
or W = U - . W
we just know that
is
contained in one of them. S
What happens if there is a germ of a complex hypersurface in V
Can we by choosing WC U-
such that
or
function,
then
is not in
U+
or
V
V
of
Po
in
Cn
= 0}=
U+
S
h
where
and in
U-.
is a holomorphic
Hence if
Po g i, then
Po
U-o A
C 2, then we can prove that and
: h(p)
is holomorphic in
On the other hand, if in
W
Po ?
W=U+?
[: = {p £ ¢n
Of course if I ~
large enough still find a neigborhood
through
is a neighborhood of
is an analytic disc in A ~ U+ ~
in
or
U-
S, S
whenever
U
is a real hypersurface is a neighborhood of
S.
THEOREM 2: Let
S
be a
C2-smooth real hypersurface in
that there is a germ exists a point Poe
~+
or
Poe
og
p e [~S ~-
a complex hypersurface such that
whenever
By the interior of
i ~ S
U
p
Cn ~
and let in
S
U
S.
through
is not in the interior of
is a connected neighborhood of
(i.e. maxdistance from
p e U
to
S).
Assume
Po"
If there
~ ~S. p
we shall mean the interior relative to
fixed sized neighborhood we shall mean that it only depends on the thickness of
Poe
U N S
Then
and [.
Po" By a
and not on
191
M A I N SECTION In this section we shall prove the m a i n theorem and simple proof of Bochners theorem on e x t e n d a b i l l t y of C-R functions on compact real h y p e r s u r f a c e s
in
C n.
Before doing the actual proof we should make a couple of observations. First we observe that if
q c ~
is a complex hyperplane (i.e. V Then there is no other germ
where
V
is the interior of
TV
of
V
VI
such that
are equal all over
manifold
for the complex structure on
(i.e. V
is of maximal dimension.)
a vectorfleld then
vINV
n
Hence
Next, we observe that itself.
So if
WI
and
such that function f u-UW I \ U -
and
q e V, C S.
Hence and
All this will imply that f
to U-
S
W2
Pl
Hence
V
Tc S
of
S
and the
is locally an integral
and it is of the same dimension as
Z E y
we can deduce that if
T c S. y
of
Z e VI,
V I = V. is
C 2 - s m o o t h so it will not turn around and touch
are two neighborhoods
f2 on U- ~ W 2 \ U-.
balls with centers
V.
S).
From the uniqueness of integral curves
h o l o m o r p h l c in say
that they are balls.
of
S
with boundary conditions
is open.
where
is a germ of a complex hyperplane in
The reason for this is that the complex tangentplane tangentplane
~ f'% S
Pl
are connected.
P2, then
and
extends h o l o m o r p h l c a l l y
Then we can by shrinking
W I~W 2
fl = f2
U-
of points
WI~ W 2 ~
P2 to
in
S
fl
on
W I and W 2 assume
If we let
WI
and
will imply that
W2
be
WI~ W 2~U-
# ~.
on W I ~ W2, so we have a h o l o m o r p h i c e x t e n s i o n
U((WlUW2)\U-).
N o w to the main part of the proof of T h e o r e m 2.
F r o m the first o b s e r v a t i o n
and the conditions on
p
given in Theorem 2 we k n o w that there is no germ of a
c o m p l e x hyperplane in
S
through
Choose a curve
y : [0,i] ÷ ~ N
F u r t h e r m o r e we want y([0,1])
v()
to
S
such that
y(O) = Po
and
to be contained in the interior of
y(1) = p. ~ ~S
and
should have finite length.
From Trepreau's 7(i) = p
p.
theorem we k n o w that there exist a neighborhood
such that all h o l o m o r p h l c functions on
U- U ( W I \ U-).
U-
or
U+
Note that we do not know from which side
say (U+
Uor
W 1 of extends U-)
the
192
functions extend. U N $
We also know that the size of
and of the shape of If
Poe
closer to
W I OS,
Po
than
Now we lift Let
S1
S
is.
off
Let
~
If not, let
¥(Pl,P)
along
Pl
be the piece of
~(Pl,P)
x([0,1>)
be a point in ~
between
p and PI"
in the following may:
be a C2-smooth surface such that
i)
SI \ WI = S \ W I
ii) i.e.
only depends on the size of
S.
then we are done. p
W1
(~(Pl,P) ~ SI) o S = Pl
~(pl,p)\{pl}
is not contained in
ill)
S I.
There exists a neighborhood V ~ S I = V N S.
iv)
V
of
¥ \Y(Pl,P)
such that
SI\ S C W I \ U - .
In other words we llft
S
off
~(Pl,P) into
W I \ U-, makes sure that
Ple
S
and
PI"
The
leaves the rest unchanged. Condition
i,li and ill
together with the first remark in this section
ensures
that there is no germ of a complex hypersurface in
size of
W I\U-
If
determines how much we can llft
SI: = {P E C n : rl(p) = O}
where
rI
S
off
SI
through
¥(pl,p).
is a C2-defining function and
U- = U ~ { p e ~n : rl(p ) < 0}, then U - ~ U- and all f ~ H(U-) extends holomori i phically to U- , i.e., H(U-) = H(U-). This last statement follows from i and iv. I I Now we can find a neighborhood
W
of
Pl
such that all functions
f ~ H(U-) i
g e H(U-)
holo-
2 extends holomorphically
to
U- ~ W 2. i
Hence all
extends to a
morphlc function in that
~ CS + I
Pl if V
U- U W . This is obtained by applying Theorem 1 and observing I 2 locally, hence not all functions holomorphic in V ~ $+ extends over i
is a small neighborhood of
Choose a point part of
¥
in
Now we llft
W2
p2~W2 qY from
SI
z2
off
~
PI"
closer to to
than
Pl
and let
Y(Pl,P2)
be the
z 3.
along
~(Pl,P2)
where: i)
Po
S 2 \ W 2 = S1 \ W 2
and get a C2-smooth hypersurface
S2
193
li)
S2
iii)
does not contain
There exists a neighborhood V ~ S2
v)
S-2
By for
$2~ S 2 C W 2 \ U ~
to
of
V
~\y(pl,P2 )
such that
. W3
of
such that all functions
P2
(UnS-) UW . 2 3
we shall mean
{p: r2(P) < 0}
r
where
is a C2-defining
function
2
S
chosen so that S- m S- = S-. 2 2 I We can keep doing this, i.e., lifting
contained
in
Wi+ I.
B(pi,r) 0 ~ C W i + I
S = {(zl,w) Say (U ~ B)
If Let
Let choose
Y(Pi-I,Pi)
Cn
until
r > o
Po
such that
to be a graph.
z I = (Zl,...,Zn_l)
if
B
Then
U-
In other words
and
w = u + iv.
is locally schlicht,
is a small neighborhood
of
pi.
in
~.
i.e.
Let us write
~(pi,R)
be a ball of radius R > o.
R
and center
In other words
B~
Pi
If
is a ball inside
~
i ~ I
then
in the
~'s local coordinates. 4~ R R q e 8[(pi , , then Bl(Pl,4--~C B[(q,2-~CBI(Pi,R ). t > 0
be small,
since
~-
If
it = {p - t(0,1)
p = -log dist(',
is Stein
q e S[(pi,~) r > 0
then
Hence
~),
P l it
e U-}
then
and
p
is plurlsubha=onic
S(pi,r)O~S~(pl,5
is still a
is a plurlsubharmonic
we want to prove that this implies
small enough then
~t
that
t > 0.
q e ~.
If we
and we can let
8~(Pi,4--~"
R Assume p e 8[(q,2--$}
q E ~l(pi,4--') but call this set
B t.
q ~ USince
and let
B~t(q,2~
8[(pi,~)CB~(q
= {p - t(O,i): and
on
for all
R contain
is
(UOB).
complex hyperplane.
u-=~-
here
S
{v < p(zl,u)}.
is a domain in
such a ball exist and sense of
off
the following way:
we may consider
is locally
instead of Let
Pl
i
: v = p (zl,u)}
U-
S
We can prove that there exists a
for each
Locally near
~-
P2"
W2 ~I
As above we obtain a neighborhood extends
the point
P2 e S2
iv)
g e H(U-)
except
Y(Pl,P2)
Wi+ I
194
Y(Pi-I,
R Pi) f ] ~ ( P i , ~ )
dist(At,~U-)
~(q)
>
# ~,
b > 0
then there exists a set
for all
t
and volume
R ~t(q,2~
At
(A t ) > a > 0
for all
By using the subaveraging principle for plurisubharmonic I ~ vol(B~) (j o t x +(-log a) vol(At)) Bt\A t Assume
vol(B t)
dist(p - t(0,i),
=
and observe
i
that if
max {0 : ~t } ~ -log(C
Hence
q ~ U-,
if
then
there exists a constant All this gives
that
functions
we get:
c > o
such that
t) = -log t + C'.
dist(q - t(0,i),
C1
t.
then locally
Hence there exists a constant
S) ~ t.
Furthermore
p e S,
such that
such that
S) = dist(q - t(O,i),
~U-).
p(q) ~ -log t - C I.
-log t ~ (l-b)(-log
If we let
t) - (log a)b + C 2.
go
t
to zero we obtain a contradiction. The reasion why can be choosen
Po
is reached at some point is the fact that
to have a finite
Y(zi,zi+l)
length.
COROLLARY: Let
M
be a complex manifold
without
bounded
a bounded domain with a C2-smooth
boundary.
compact
and all
K c ~ ~
holomorphic
such that
function
in
D C~
complex hypersurfaces.
Then there exist a domain C-R
functions
on
~D
is
If D
~
extends
and a to a
~ \K.
PROOF: First we observe hypersurface
~
that there is no bounded domain in
in its boundary.
the interior with respect is a point
p e ~ 3D
This implies W(z)
of
W(z)
where
extend.
z
to
Hence,
~]
so that
p
that whenever
Since
~D
is compact,
containing
if there is a germ ~
is not contained
~D
in
int~
a complex
int(~ ~ ~D)
of a complex hypersurface
in
[this is
~D, then there
is compact.
z g ~D, then there is a fixed sized neighborhood
such that C-R functions W(z): = {W(z) \ D
M
or
on
~D
W(z)OD}
does have a holomorphic depending
extension
on to which side they
then we may assume that the size of
W(z)
does
to
195
have a minimum. Let extension
R(D): = to
Define
U W(z), ze~D R(D).
then all C-R functions
~: = int(R(D)<jD)
and
on
K: = D k R ( D ) .
~D
Then
does have a holomorphlc
DC~
and
R(D) = ~ k K .
REFERENCES
I.
J.M. Tr6preau: "Sur le prolongement holomorphe une hypersurface reelle de classe C 2 dans ~n.(1986).
des functions C-R definles sur Invent. Math. 83, 583-592
