Real Submanifolds in Complex Space and Their Mappings (PMS-47) 9781400883967

Table of contents :
CONTENTS
Preface
Chapter I. Hypersurfaces and Generic Submanifolds in C^N
§1.1. Real Hypersurfaces in C^N
§1.2. Holomorphic and Antiholomorphic Vectors
§1.3. CR, Totally Real, and Generic Submanifolds
§1.4. CR Vector Fields and CR Functions
§1.5. Finite Type and Minimality Conditions
§1.6. Coordinate Representations for CR Vector Fields
§1.7. Holomorphic Extension of CR Functions
§1.8. Local Coordinates for CR Manifolds
Chapter II. Abstract and Embedded CR Structures
§2.1. Formally Integrable Structures on Manifolds
§2.2. Levi Form and Levi Map of an Abstract CR Manifold
§2.3. CR Mappings
§2.4. Approximation Theorem for Continuous Solutions
§2.5. Further Approximation Results
Chapter III. Vector Fields: Commutators, Orbits, and Homogeneity
§3.1. Nagano’s Theorem
§3.2. Sussman’s Theorem
§3.3. Local Orbits of Real-analytic Vector Fields
§3.4. Canonical Forms for Real Vector Fields of Finite Type
§3.5. Canonical Forms for Real Vector Fields of Infinite Type
§3.6. Weighted Homogeneous Real Vector Fields
Chapter IV. Coordinates for Generic Submanifolds
§4.1. CR Orbits, Minimality, and Finite Type
§4.2. Normal Coordinates for Generic Submanifolds
§4.3. Canonical Coordinates for Generic Submanifolds
§4.4. Weighted Homogeneous Generic Submanifolds
§4.5. Normal Canonical Coordinates
Chapter V. Rings of Power Series and Polynomial Equations
§5.1. Finite Codimensional Ideals of Power Series Rings
§5.2. Analytic Subvarieties
§5.3. Weierstrass Preparation Theorem and Consequences
§5.4. Algebraic Functions, Manifolds, and Varieties
§5.5. Roots of Polynomial Equations with Holomorphic Coefficients
Chapter VI. Geometry of Analytic Discs
§6.1. Hilbert and Poisson Transforms on the Unit Circle
§6.2. Analytic Discs Attached to a Generic Submanifold
§6.3. Submanifolds of a Banach Space
§6.4. Mappings of the Banach Space C^1.α
§6.5. Banach Submanifolds of Analytic Discs
Chapter VII. Boundary Values of Holomorphic Functions in Wedges
§7.1. Wedges with Generic Edges in C^N
§7.2. Holomorphic Functions of Slow Growth in Wedges
§7.3. Continuity of Boundary Values
§7.4. Uniqueness of Boundary Values
§7.5. Additional Smoothness up to the Edge
§7.6. Further Results and an “Edge-of-the-Wedge” Theorem
Chapter VIII. Holomorphic Extension of CR Functions
§8.1. Criteria for Wedge Extendability of CR Functions
§8.2. Sufficient Conditions for Filling Open Sets with Discs
§8.3. Tangent Space to the Manifold of Discs
§8.4. Defect of an Analytic Disc Attached to a Manifold
§8.5. Ranks of the Evaluation and Derivative Maps
§8.6. Minimality and Extension of CR Functions
§8.7. Necessity of Minimality for Holomorphic Extension to a Wedge
§8.8. Further Results on Wedge Extendability of CR Functions
Chapter IX. Holomorphic Extension of Mappings of Hypersurfaces
§9.1. Reflection Principle in the Complex Plane
§9.2. Reflection Principle: Preliminaries
§9.3. Reflection Principle for Levi Nondegenerate Hypersurfaces
§9.4. Essential Finiteness for Real-analytic Hypersurfaces
§9.5. Formal Power Series of CR Mappings
§9.6. Reflection Principle for Essentially Finite Hypersurfaces
§9.7. Polynomial Equations for Components of a Mapping
§9.8. End of Proof of the Reflection Principle
§9.9. Reflection Principle for CR Mappings
§9.10. Reflection Principle for Bounded Domains
§9.11. Further Results on the Reflection Principle
Chapter X. Segre Sets
§10.1. Complexification of a Generic Real-analytic Submanifold
§10.2. Definition of the Segre Manifolds and Segre Sets
§10.3. Examples of Segre Sets and Segre Manifolds
§10.4. Basic Properties of the Segre Sets
§10.5. Segre Sets, CR Orbits, and Minimality
§10.6. Homogeneous Submanifolds of CR Dimension One
§10.7. Proof of Theorem 10.5.2
Chapter XI. Nondegeneracy Conditions for Manifolds
§11.1. Finite Nondegeneracy of Abstract CR Manifolds
§11.2. Finite Nondegeneracy of Generic Submanifolds of C N
§11.3. Holomorphic Nondegeneracy
§11.4. Essential Finiteness for Real-analytic Submanifolds
§11.5. Comparison of Nondegeneracy Conditions
§11.6. Compact Real-analytic Generic Submanifolds
§11.7. Nondegeneracy for Smooth Generic Submanifolds
§11.8. Essential Finiteness of Smooth Generic Submanifolds
Chapter XII. Holomorphic Mappings of Submanifolds
§12.1. Jet Spaces and Jets of Holomorphic Mappings
§12.2. Basic Identity for Holomorphic Mappings
§12.3. Determination of Holomorphic Mappings by Finite Jets
§12.4. Infinitesimal CR Automorphisms
§12.5. Finite Dimensionality of Infinitesimal CR Automorphisms
§12.6. Iterations of the Basic Identity
§12.7. Analytic Dependence of Mappings on Jets
Chapter XIII. Mappings of Real-algebraic Subvarieties
§13.1. Mappings between Generic Real-algebraic Submanifolds
§13.2. Some Necessary Conditions for Algebraicity of Mappings
§13.3. Mappings of Real-algebraic Subvarieties
References
Index

Citation preview

R eal Subm anifolds in Complex Space and Their M appings

Princeton Mathematical Series E d it o r s : J o h n

N.

M a th e r

and E l i a s

M . S t e in

1. The Classical Groups by Hermann Weyl 4. Dimension Theory by W. Hurewicz and H. Wallman 8. Theory o f Lie Groups: I by C. Chevalley 9. 14. 17. 19. 25. 28. 30. 31. 32. 33. 34. 35.

Mathematical Methods o f Statistics by H arold Cramer The Topology o f Fibre Bundles by Norman Steenrod Introduction to Mathematical Logic, Vol. I by Alonzo Church Homological Algebra by H. Cartan and S. Eilenberg Continuous Geometry by John von Neumann Convex Analysis by R. T. Rockafellar Singular Integrals and Differentiability Properties o f Functions by E. M. Stein Problems in Analysis edited by R. C. Gunning Introduction to Fourier Analysis on Euclidean Spaces by E. M. Stein and G. Weiss Etale Cohomology by J. S. Milne Pseudodifferential Operators by Michael E. Taylor Three-Dimensional Geometry and Topology: Volume 1 by William P. Thurston. Edited by Silvio Levy 36. Representation Theory o f Semisimple Groups: An Overview Based on Examples by Anthony W. Knapp 37. Foundations o f Algebraic Analysis by Masaki Kashiwara, Takahiro Kawai, and Tatsuo Kimura. Translated by Goro Kato 38. Spin Geometry by H. Blaine Lawson, Jr., and Marie-Louise Michelsohn 39. Topology o f 4-Manifolds by Michael H. Freedman and Frank Quinn 40. Hypo-Analytic Structures: Local Theory by Francois Treves 41. The Global Nonlinear Stability o f the Minkowski Space by Demetrios Christodoulou and Sergiu Klainerman 42. Essays on Fourier Analysis in Honor o f Elias M. Stein edited by C. Fefferman, R. Fefferman, and S. Wainger 43. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals by Elias M. Stein 44. Topics in Ergodic Theory by Ya. G. Sinai 45. Cohomological Induction and Unitary Representations by Anthony W. Knapp and D avid A. Vogan, Jr. 46. Abelian Varieties with Complex Multiplication and Modular Functions by Goro Shimura 47. Real Submanifolds in Complex Space and Their Mappings by M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild

REAL SUBMANIFOLDS IN COM PLEX SPACE AND THEIR MAPPINGS

M. Salah Baouendi Peter Ebenfelt Linda Preiss Rothschild

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

Copyright © 1999 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved Baouendi, M. Salah, 1937Real submanifolds in complex space and their mappings / M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild, p. cm.— (Princeton mathematical series ; 47) Includes bibliographical references and index. ISBN 0-691-00498-6 (cl. : alk. paper) 1. Submanifolds. 2. Functions of several complex variables. 3. Holomorphic mappings. I. Ebenfelt, Peter. II. Rothschild, Linda Preiss, 1945-. III. Title. IV. Series. QA649.B36 1998 516.362—dc21 98-44235

ISBN 0-691-00498-6

The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed The paper used in this publication meets the minimum requirements of ANSI/NISO Z39.48-1992 (R1997) (Permanence o f Paper) http://pup.princeton.edu

3 5 7 9

10 8 6 4 2

to Moungi and Meriem Ann-Sofie, Felicia, and Evelina David and Daniel

CONTENTS

xi

Preface Chapter §1.1. § 1.2. §1.3. § 1.4. § 1.5. §1.6. § 1.7. § 1.8.

I. Hypersurfaces and1Gen Generic Submanifolds in cN Real Hypersurfaces in C N Holomorphic and Antiholomorphic Vectors fenericSubmanifolds Submanifolds CR, Totally Real, and Generic Functions CR Vector Fields and CRRFunctions ilityConditions Conditions Finite Type and Minimality ionsfor forCR CRVector VectorFields Fields Coordinate Representations CRFunctions Functions Holomorphic Extension iofofCR Local Coordinates for CR Manifolds

3 3 6 9 14 17 21 26 30

Chapter II. Abstract and Embedded CR Structures §2.1. Formally Integrable Structures on Manifolds §2.2. Levi Form and Levi Map of an Abstract CR Manifold §2.3. CR Mappings §2.4. Approximation Theorem for Continuous Solutions §2.5. Further Approximation Results

35 35 40 49 52 57

Chapter §3.1. §3.2. §3.3. §3.4. §3.5. §3.6.

III. Vector Fields: Commutators, Orbits, and Homogeneity Nagano’s Theorem Sussman’s Theorem Local Orbits of Real-analytic Vector Fields Canonical Forms for Real Vector Fields of Finite Type Canonical Forms for Real Vector Fields of Infinite Type Weighted Homogeneous Real Vector Fields

62 62 68 73 73 87 91

Chapter §4.1. §4.2. §4.3. §4.4. §4.5.

IV. Coordinates for Generic Submanifolds CR Orbits, Minimality, and Finite Type Normal Coordinates for Generic Submanifolds Canonical Coordinates for Generic Submanifolds Weighted Homogeneous Generic Submanifolds Normal Canonical Coordinates vii

94 94 95 101 108 112

viii

CONTENTS

Sauations Chapter V.Rings of Power Series and Polynomial Equations §5.1. Finite Codimensional Ideals of Power Series Rings §5.2. Analytic Subvarieties §5.3. Weierstrass Preparation Theorem and Consequences §5.4. Algebraic Functions, Manifolds, and Varieties §5.5. Roots of Polynomial Equations with Holomorphic Coefficients

119 119 128 132 139 145

Chapter VI. GeometryGeometry of Analytic Discs §6.1. Hilbert and Poisson Transforms on the Unit Circle §6.2. Analytic Discs Attached to a Generic Submanifold §6.3. Submanifolds of a Banach Space §6.4. Mappings of the Banach Space C La §6.5. Banach Submanifolds of Analytic Discs

156 156 162 166 176 178

Chapter VII. Boundary Values of Holomorphic Functions in Wedges 184 §7.1. Wedges with Generic Edges in i C N 184 iwGrowth G row th in in Wedges W erises 185 §7.2. Holomorphic Functions of Slow §7.3. Continuity of Boundary Values 192 §7.4. Uniqueness of Boundary Values 196 §7.5. Additional Smoothness up to the Edge 202 §7.6. Further Results and an “Edge-of-the-Wedge” Theorem 204 Chapter VIII. Hold Holomorphic Extension of CR Functions 205 §8.1. Criteria for Wedge Extendability of CR Functions 205 §8.2. Sufficient Conditions for Filling Open Sets with Discs 206 §8.3. Tangent Space to the Manifold of Discs 212 §8.4. Defect of an Analytic Disc Attached to a Manifold 218 §8.5. Ranks of the Evaluation and Derivative Maps 224 §8.6. Minimality and Extension of CR Functions 230 §8.7. Necessity of Minimality for Holomorphic Extension to a Wedge 231 §8.8. Further Results on Wedge Extendability of CR Functions 238 Chapter §9.1. §9.2. §9.3. §9.4. §9.5. §9.6. §9.7. §9.8. §9.9.

IX.Holomorphic Extension of Mappings of Hypersurfaces irfaces 241241 Reflection Principle in the Complex Plane 242 Reflection Principle: Preliminaries 243 Reflection Principle for Levi Nondegenerate Hypersurfaces 246 Essential Finiteness for Real-analytic Hypersurfaces 248 Formal Power Series of CR Mappings 252 Reflection Principle for Essentially Finite Hypersurfaces 255 Polynomial Equations for Components of a Mapping 257 End of Proof of the Reflection Principle 259 Reflection Principle for CR Mappings 265

CONTENTS

§9.10. §9.11.

Reflection Principle for Bounded Domains Further Results on the Reflection Principle

ix

270 277

Chapter X. Segre Sets § 10.1. Complexification of a Generic Real-analytic Submanifold §10.2. Definition of the Segre Manifolds and Segre Sets § 10.3. Examples of Segre Sets and Segre Manifolds § 10.4. Basic Properties of the Segre Sets § 10.5. Segre Sets, CR Orbits, and Minimality § 10.6. Homogeneous Submanifolds of CR Dimension One §10.7. Proof Proof of Theorem 10.5.2

281 281 283 289 293 300 305 312

Chapter XI. NNondegeneracy Conditions for Manifolds §11.1. Finite Nondegeneracy of Abstract CR Manifolds §11.2. Finite Nondegeneracy of Generic Submanifolds of) f c w §11.3. Holomorphic Nondegeneracy § 11.4. Essential Finiteness for Real-analytic Submanifolds § 11.5. Comparison of Nondegeneracy Conditions §11.6. Compact Real-analytic Generic Submanifolds § 11.7. Nondegeneracy for Smooth Generic Submanifolds §11.8. Essential Finiteness of Smooth Generic Submanifolds

315 315 319 322 325 329 335 336 342

Chapter XII. Ho Holomorphic Mappings of Submanifolds §12.1. Jet Spaces and Jets of Holomorphic Mappings § 12.2. Basic Identity for Holomorphic Mappings § 12.3. Determination of Holomorphic Mappings by Finite Jets §12.4. Infinitesimal CR Automorphisms §12.5. Finite Dimensionality of Infinitesimal CR Automorphisms § 12.6. Iterations of the Basic Identity § 12.7. Analytic Dependence of Mappings on Jets

349 349 352 358 361 366 370 373

Chapter XIII. Mappings of Real-algebraic Subvarieties §13.1. Mappings between Generic Real-algebraic Submanifolds § 13.2. Some Necessary Conditions for Algebraicity of Mappings § 13.3. Mappings of Real-algebraic Subvarieties

379 379 383 387

References

390

Index

401

PREFACE

The study of real submanifolds in C N has emerged as an important area in several complex variables and has gained independent interest in the past 40 years. In fact, some of the problems studied here go back to H. Poincar£ and E. Cartan in the early 1900’s. The “modern era” in this subject dates back to the 1950’s with the fundamental contributions of H. Lewy, who discovered deep connections between the theory of several complex variables and that of partial differential equations. In this book we focus on a number of important problems for which substantial progress has been made in the last twenty years, such as holomorphic extendibility of Cauchy-Riemann (CR) functions from real submanifolds, holomorphic extendibility of mappings between such manifolds, and the structure of the germs of biholomorphisms which map one real submanifold into another. The case of real algebraic submanifolds is of particular interest and is also addressed in this book. We begin by developing the tools necessary to present complete, self-contained results in the areas mentioned above. These tools come from a variety of fields, including geometry, analysis, commutative algebra, as well as more classical com­ plex analysis. We have tried to include background material which cannot be easily found in other books or monographs. In the first two chapters we introduce some basic definitions such as that of CR manifolds, CR vector fields, Levi form, finite type and minimality conditions, and prove some basic properties of these objects. Chapter III deals with real vector fields; we prove the theorems of Nagano and of Sussman concerning orbits of such vector fields, and also establish the existence of weighted homogeneous coordinates. These results are used in Chapter IV to give various kinds of canonical coordinates for embedded submanifolds in . Chapter V, which is independent of the previous chapters, develops mostly algebraic methods for working with rings of formal and convergent power series and polynomials. In Chapter VI we study boundary values of holomorphic functions with slow growth, defined in a wedge whose edge is a generic submanifold in C ^, and prove some uniqueness and regularity results. In Chapter VII we develop the theory of analytic discs attached to a generic submanifold of and show that the set of such (small) discs forms an infinite dimensional Banach submanifold in the space of all discs. The Bishop equation is xi

xii

PREFACE

treated here in a Banach space setting. This approach is then used in Chapter VIII to prove the theorem of Tumanov showing that minimality implies holomorphic extendibility of all CR functions into an open wedge in C N. The necessity of this condition is also given. Another main topic of this book is addressed in Chapter IX. A reflection principle for CR mappings between real analytic hypersurfaces is proved here, and applications to proper holomorphic mappings between bounded domains in C N are given. In Chapter X we develop the theory of Segre sets for generic submanifolds. An equivalent condition for minimality in terms of Segre sets, which will be an essential tool for the remainder of the book, is established here. In Chapter XI we introduce a number of nondegeneracy conditions for real submanifolds, including holomorphic nondegeneracy and essential finiteness, and explore the relationships among them. In the last two chapters we study germs of holomorphic mappings which send one real analytic (or algebraic) submanifold into another and show, under some nondegeneracy conditions, that such germs are determined by their jets of finite order. In the algebraic case we give sufficient conditions for all such germs to be algebraic mappings. The material in this book is intended to be accessible to mature graduate stu­ dents; no previous knowledge of several complex variables is assumed of the reader. Although most of the results presented could be found in research articles, we have also included some previously unpublished work. There are many important areas of several complex variables connected with the study of real submanifolds which are not addressed in this book. Most notable is the omission of the Kohn 3-Neumann problem and its many consequences. We also do not include any discussion of the Chern-Moser normal forms for Levinondegenerate hypersurfaces. We have been greatly inspired by the work of many mathematicians who have made fundamental contributions to the topics covered in this book. We will not attempt to list them all here. At the end of each chapter we have included notes giving bibliographical references and historical comments, and we apologize for any errors or sins of omission. Finally, we would like to make special mention here of some of our collaborators in this field over the years, including Steven Bell, Xiaojun Huang, Howard Jacobowitz, Elias M. Stein, Frangois Treves, and Jean-Marie Trepreau, from whom we learned a great deal. We would also like to thank Dmitri Zaitsev for numerous useful comments on some chapters of this book. M. S a l a h B a o u e n d i P L

July 1998

in d a

P

eter

r e is s

R

E

ben felt

o t h s c h il d

Real Subm anifolds in Complex Space and Their M appings

CHAPTER I

HYPERSURFACES AND GENERIC SUBMANIFOLDS IN C N

Summary The basic object of study in this book is a smooth real submanifold in C N whose tangent space has a smoothly varying maximal complex subspace. Here, there is a rich interaction between real and complex function theory as well as geometry. In this chapter, we give basic definitions and properties of such objects. We begin with a brief discussion of real hypersurfaces. In §1.2, we define holomorphic and antiholomorphic vectors for real submanifolds in C^. These play a crucial role throughout this book. We then define the notions of CR, generic, and totally real submanifolds of C ^, and introduce the CR vector fields and CR functions on these submanifolds. The important notions of minimality and finite type are defined in §1.5; these are discussed in terms of coordinates for hypersurfaces in §1.6. A brief discussion of holomorphic extension of CR functions on real-analytic generic submanifolds and formal holomorphic power series of CR functions is given in § 1.7. The chapter concludes with a description of local coordinates for general CR submanifolds and the intrinsic complexification for real-analytic CR submanifolds. §1.1. Real Hypersurfaces in We begin with some notation which will be used throughout this book. For Z e C N we write Z = ( Z j , . .. ,Zyv), where Z; = xj -f iyj, with Xj and yj real numbers; we write Z = ( Z j , ... , Z#), where Zj = Xj — iy; , the complex conjugate of Z, . As is customary, we use i to denote the imaginary unit \/^ T . The absolute value of Z, is given by |Z712 = x j -F yj. We identify with R2/v, and denote a function / on a subset of as /(jc, y), or, by abuse of notation, as /(Z ,Z ). A (smooth) real hypersurface in C N is a subset M of C N such that for every point po € M there is a neighborhood U of po in and a smooth real-valued function p defined in U such that (1.1.1)

M H U = {Z e U : p(Z , Z) = 0},

with differential dp nonvanishing in U> i.e. at every point p of U not all first derivatives of p vanish at p. Such a function p is called a local defining function 3

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN C*

4

for M near po. The hypersurface M is real-analytic if the defining function p in ( 1.1.1) can be chosen to be real-analytic. Example 1.1.2. The hypersurface given by the equation Im Z# = 0 is a “flat” hyperplane in CN. E xample 1.1.3. The hypersurface in CN given by the equation TV—1

Im Z „ = £ | z / ;= i is called the Lewy hypersurface. given by \%j\2 = 1 is a com­ pact hypersurface. The reader can check that the holomorphic rational mapping H (Z ) = (H \( Z ) ,. .. , Hn (Z)) given by Example 1.1.4. The unit sphere in

iZ j H j(z ) : = - :---- -}=~,

j =

Hn ( Z) :=

i(Zhi + 1)

takes the unit sphere minus the point (0, 0 , . . . ,1 ) bijectively to the Lewy hyper­ surface given in Example 1.1.3. R em ark 1.1.5. For any hypersurface M and any p 0 € M there exist smooth coordinates (jcJ, . . . , x ’2N) near po in K2N, vanishing at po, such that M is given by x\ = 0 in a neighborhood of po. Indeed, after an affine change of coordinates we may assume p0 = 0 and 3p/dxi(0) ^ 0. Then setting x[ = p, x fj = jc7, 2 < j < N , and x ’k+N = 1 < Jc < N, gives the desired conclusion. Hence all real hypersurfaces are locally equivalent after a smooth change of coordinates. However, in general there is no holomorphic change of coordinates which performs this equivalence.

The reader can easily check that if p and p' are two defining functions for M near po, then there is a nonvanishing real-valued smooth function a defined in a neighborhood of po such that p = ap' near po. P ro p o sitio n 1.1.6. Let M be a real hypersurface in C /l+1 through po- Then there are holomorphic coordinates (z, w) near po, vanishing at po, z € C '\ w = s + it G C, and a real-valued smooth function (p(z,z,s) defined near 0 in R 2/i+i with 0(0) = 0, dz)j.k = dj/dzic, s is the d x d matrix given by (&),-.* = 3j/dsk and

§1.6. COORDINATE REPRESENTATIONS FOR CR VECTOR FIELDS

23

We leave the details of the calculation to the reader. In later chapters, we shall abuse the notation and use L \ , . . . , L n to denote the CR vector fields of M in any set of local coordinates. It is worth noting separately the case where M is a hypersurface. Recall the form of the defining equation (1.1.7) for M in regular coordinates (z, w) given by Proposition 1.1.6. Proposition 1.6.4. I f M isa hypersurface in C N given by (1.1.7), then the basis (1.6.2) o f the CR vectorfields on M is given by

(1.6.5)

a

Lj = — dZj

fa,

a

Aj = — OZj

j = I , . . . ,n.

— , 1 4* l(/>s dw

In the parametrization *P : (z, z, s) h* 'P -1 is given by ( 1.6.6)

a

0z,

(z ,

s + i(j)(z,z, s)), their pushforward by

a

j = \,...,n.

1 -b l OZj OZk

t

—2.

P r o o f . To prove (1.6.10), we note first that, by Remark 1.5.8, [A ; , A *] is aC R vector field and hence a linear combination of the A p with smooth coefficients. On the other hand, from the form of (1.6.6), [ A y , A * ] is a multiple of Jj, so that the multiple must be 0. The form of (1.6.6) also gives (1.6.11) and (1.6.12) immediately. To prove (1.6.14), we use a direct calculation to show

(1.6.16) 2ip,k = i - ^ f - + ls V 1 + * '& / /

.

On the other hand, by (1.6.13) we have,

(1.6.17)

fc,*,(z,z. 0) = J - J L Pr(z ,z) + 0 ( \ z r l ) = 0 ( | z r 2), dZk d Z j

and similarly, (1.6.18)

4>z,(z,z, 0) = 0 (|z|r '), Zk(z, z, 0) = 0 (|z|r '), 4>siz, z, 0) = O(lzl).

Putting s = 0 in (1.6.16) and using (1.6.17) and (1.6.18) yield (1.6.14). In order to prove (1.6.15) we use an induction on q , beginning with q = 0, for which the

§1.6. COORDINATE REPRESENTATIONS FOR CR VECTOR FIELDS

25

result is given by (1.6.14). The details are left to the reader. This completes the proof of Proposition 1.6.9. □ The following is also needed for the proof of Theorem 1.6.7. P roposition 1.6.19. Given a hypersurface M C Cn+1, po e M, and a positive integer J > 1, there exist regular coordinates (z, w) near po, vanishing at po, such that M is given locally by ( 1.1.7) with

(1.6.20)

(z, z, 0) = qiz, z) + 0 (|z |'+l),

where q( z , z ) is a real polynomial (not necessarily homogeneous) o f total degree J satisfying q(z, 0) == q(0, z) = 0. P roof. We prove the proposition by induction on / . By the definition of regular coordinates, we may choose (z, tv) so that (1.6.20) is satisfied for J = 1 (in which case q( z , z ) = 0). Assume (1.6.20) for J replaced by J — 1; we shall prove it for 7. Writing the Taylor expansion of 0(z, z, 0) at the origin we have 0 (z,z,O) = t ( z , z ) + 0 (\z\J+l), where £(z,z) is a polynomial of total degree J. Using the induction hypothesis, we have £ ( z , z ) = I m h j ( z ) + q ( z , z ), where hj is a homogeneous, holomorphic polynomial of degree J and q(z, z) a real polynomial (not necessarily homogeneous) of total degree J satisfying q(z, 0) == q(0, z) = 0. We consider the holomorphic change of coordinates defined by z! = z, w f = w — hj(z). Now M is defined by Im wf = 0'(z', z', s'), where

0'(z', z \ s') = (z\ z', s' + Re hj(z' )) - Im h j ( z '),

s' = Re w'.

It follows from the assumption on 0 that 0. For this, we let A i , . . . , A n be the basis for the CR vector fields on M given by (1.6.2'). Observe that we have, for all a, ft, y, and j = 1, . . . , n, AjZa = A j ( s + /0(z, z, s ) ) fi 5= 0,

(1.7.19)

Aj} ? = PjZfi[j\

where p[j] = ( P\ , . . . , fij — 1, . . . , fin). We shall use the notation A^ = A ^ . . . A„n for any multi-index /3. To show that dapY = 0, for \f}\ >0, we apply A^ to each term in the Taylor series (1.7.18) of h( z, z, s ) and set z. = 0. This completes the proof in view of (1.7.19), since A = 0 for every p with |^ | > 0 .

□

§1.8. Local Coordinates for CR Manifolds In this section we give a convenient system of local coordinates for arbitrary CR submanifolds and we prove that any smooth CR submanifold is locally a graph of a CR mapping over a generic submanifold in a lower dimensional space. We also show that any real analytic CR submanifold can be embedded as a generic submanifold of some complex submanifold. We have the following. T heorem 1.8.1. Let M be a CR submanifold o f codimension d in C N through

po. Then there exist an integer d\, holomorphic coordinates near po, van­ ishing at po, a generic manifold M\ in C N~d+d\ and d — d\ CR functions = W u • • • i i^d-dx) on M i, 0(0) = 0, such that, near po, M is the graph o fty over M\, i.e. ( 1. 8 .2)

M = {(m, 0 ( m ) ) e C N : m e M\}.

§1.8. LOCAL COORDINATES FOR CR MANIFOLDS

31

If M is real analytic, then the above holomorphic coordinates can be chosen so that x/s =0. Proof. Let p i , ... , Pd be defining functions for M in C N, and d\ the rank of {3pi,... , dpd}. We now proceed as in the proofs of Propositions 1.3.6 and 1.3.8. By linear algebra and an application of the implicit function theorem we may choose coordinates (Zl

Z N - d t , w ‘1, . . . , w'2di_ d , w'{, . . . , Wd _ dx),

vanishing at po, so that after a linear transformation the py have the form Pj — Im Wj —< p j ( z , z, Re u/),

1 < j < 2d\ — d,

(1.8.3)pj+2d\ —d = Re (wj - x/fjiz, z , Re u/)), 1 < j < d -

du

Pj+dl = Im (wj - ilfj(z, z , Re w'))9 1 < j < d - du where the 0y are smooth, real valued functions, and the x/rj are smooth complex valued, all vanishing at the origin together with their differentials. Let M\ be the generic manifold in C N-(d-do define(i near pQ\>y (1.8.4)

Im w'j —j(z, z , Re w f) = 0, 1 < j < 2d\ —d.

Denote by V and Vi the CR bundles of M and M\ respectively. Note that dimcV = dimcVi = N — d\. It remains to show that the x/fj are CR functions on M \. Let £ y, 1 < j < N - d \, be antiholomorphic vector fields in C N~(d~dl) such that their restrictions to M\ form a local basis of the sections of Vi near po = 0. Since (1.8.5)

jCjPfc(p) = 0, p e Mi, l < j < N —d\,

\ < k < 2 d \ —d,

any section L of V , i.e. L e T(M, V), is necessarily of the form d-d{ (1.8.6)

L = E 7=1

^

~

N-di 7 + E ^ -

w j

y=l

From the equations (1.8.7)

L(pj+2d\ —d + ipj+d{) = 0,

1 < j < d —d\,

and using (1.8.3) and (1.8.6), it follows that (1.8.8)

( J 2 d j C j ) ^ = 0, k = l , . . . , d - d u 7=1

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN C N

32

holds on M. Applying L to pj+2dx-d* 1 5 7 < d — d\, and using the fact that the differentials of the xj/j vanish at 0, it follows that the functions Xj in (1.8.6) satisfy Xj(0) == 0, j = 1, . . . d —d\. Hence we may choose a basis for the sections of V of the form (1.8.9)

L

+ M

dWk

where the Xj are smooth functions near 0. It then follows from (1.8.8) that the functions xj/k are annihilated by the £ , . Since the Cj, j = 1, . . . , N — d\, span the CR vector fields on M\ near po, it follows that the xj/k are CR functions on M \ . This proves (1.8.2) with xj/ a CR vector-valued function on M\. It remains to show that if M is real-analytic then one can choose the xj/ in (1.8.2) to be zero. For this, we note first that if M is real-analytic the xj/j in (1.8.3) are real-analytic. By Corollary 1.7.13, since the xj/j are also CR functions on M \, they extend to be holomorphic in some neighborhood of M\ in . Let Hj (z, it/) be the holomorphic extension of x//j(z, z, u/). Then if the Wj are replaced by the new holomorphic functions Wj — H j ( z , u/), j = 1,.. .d — d\, we achieve the desired form, i.e. xj/ = 0 in (1.8.2). □ As an immediate corollary of Theorem 1.8.1, we obtain the important result that a real-analytic CR submanifold is locally a generic submanifold in a (possibly) lower dimensional complex space. C o r o l la r y 1.8.10. Let M be a real-analytic CR submanifold in C N through po. Then there exist holomorphic coordinates (Z', Z") e Cm' x Cm", with m ’ + m" = N, near po and vanishing at po, such that M C {Z " = 0} = C m near po with M a generic submanifold o f {Z " = 0} = C m .

- roof d lt . By Theorem 1.8.1, there are local coordinates (z, w \ w") near po, P vanishing at po, so that M is given by Im w'j - 0y(z, z, Re u/), j = 1, . . . 2d\ - d, w” = 0. If we take Z ' = (z, w') and Z" = w", the result immediately follows.

□

The following result shows that a generic submanifold is not contained in any proper complex submanifold. lP roposition < j < N 1.8.11. Let M c C N be a smooth CR submanifold. I f M is real-analytic, then the following two conditions are equivalent. (i) M is a generic submanifold o f C N. (ii) Any germ o f a holomorphic function at any point in M which vanishes on M must vanish identically.

NOTES FOR CHAPTER I

33

I f M is assumed to be only smooth, then (i) implies (ii). P roof. The fact that if M is smooth then (i) implies (ii) is a consequence of

Proposition 1.3.11. Indeed, through any point po € Af, there exists a maximally totally real submanifold of contained in M, as can be seen by taking regular coordinates (z, w) vanishing at po (Proposition 1.3.6) and restricting to Im z = 0 in (1.3.7). Now suppose M is real-analytic. We shall show that if M is not generic, then (ii) fails to hold. If M is not generic, there is a point po e M around which M is not generic. By Corollary 1.8.10, since M is CR, we can find holomorphic coordinates Z = (Z', Z") in Cm' x Cm" with m" > 0, such that near /?o> M is contained in {Z" = 0}. Hence the components of Z" are holomorphic functions which vanish identically on Af contradicting (ii). The proof of Proposition 1.8.11 is complete. □ Using Proposition 1.8.11, one can check that the submanifold given by Z" = Ois the unique holomorphic submanifold of C N of smallest possible dimension which contains M as a generic submanifold. Indeed, if h is a holomorphic function which vanishes on M, then by Proposition 1.8.11, the restriction of h to Z" = 0 vanishes identically. Hence any local holomorphic submanifold containing AT must also contain the submanifold given by Z" = 0, i.e. the latter is the smallest germ of holomorphic submanifold containing M. This motivates the following definition. DEfiNiTiON 1.8.12. Let Af be a real-analytic CR submanifold in C N through Pq. Then the germ at po of the holomorphic submanifold of of smallest possible dimension containing the germ of M at po is called the intrinsic complexification of Af at po. R emark 1.8.13. By making use of Corollary 1.8 . 10, many questions concern­ ing local properties of real-analytic CR manifolds can be reduced to the case of generic submanifolds.

Notes for Chapter I The study of real hypersurfaces in complex space goes back to Poincar6 [1] and Cartan [1], [2] early in the twentieth century. It was Lewy [1], [2] who discovered the importance of the CR vector fields of a real hypersurface in connection with his study of boundary values of holomorphic functions. This led to his example of a first order partial differential equation without solution. The notion of finite type was introduced in PDE theory by Hormander [1]. Finite type in several complex variables, as defined in this chapter, was studied by Kohn [1] for hypersurfaces and later by Bloom and Graham [1] for higher codimensional generic submanifolds. Minimality was defined by Tumanov [1] in connection with the extension of CR

34

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN C N

functions into wedges (see also Chapter VII and its notes) and earlier by Trepreau [1] for the case of a hypersurface. Most of the other definitions and results in this chapter are by now standard. We mention some early works where these concepts appeared, including Rossi [1], Tomassini [1], Greenfield [1], Andreotti-Hill [1]. It would be difficult to pinpoint to whom each definition and/or result is due. In this book we have restricted attention to CR manifolds, i.e. those for which the CR dimension is constant. However, important results regarding submanifolds for which the CR dimension varies from point to point have been obtained by a number of mathematicians, including Bishop [1], Moser and Webster [1], Kenig and Webster [1]. For basic material dealing with real submanifolds in complex space, we mention the recent books by Boggess [1], D’Angelo [3], Jacobowitz [1], and Treves [2]. The reader can consult e.g. the books by Helgason [1] or Kobayashi and Nomizu [1] for standard material on differential geometry.

CHAPTER II

ABSTRACT AND EMBEDDED CR STRUCTURES

Summary In this chapter, we define formally integrable and integrable structures on differ­ entiable manifolds. Abstract CR manifolds are a special case of these structures. In §2.2 we define the Levi form and Levi map of an abstract CR manifold and give some basic properties. We also define pseudoconvexity for CR manifolds of hypersurface type and prove some basic results for embedded hypersurfaces. A discussion of CR mappings between CR manifolds is given in §2.3. In §2.4 and §2.5 we prove an approximation theorem for solutions of integrable structures. This result, which implies that CR functions on generic submanifolds of C N can be uniformly approximated by holomorphic polynomials, will be crucial in Chapter VIII, where holomorphic extension of CR functions is studied. §2.1. Formally Integrable Structures on Manifolds In this section we consider a subbundle V of C T M , the complexified tangent bundle of M, where M is a smooth manifold. (Here we do not assume that M is embedded as a submanifold of C ^, as in Chapter I.) For p e M , let n be the complex dimension of the fiber Vp of V at p and write dim^ M = n + m. We shall say that V is formally integrable (or involutive) if the space of smooth sections of V, C°°(M, W ) , is closed under commutators, i.e. if L \ , L2 e C°°(M, W ) , then [Lj, L 2] € C°°(M, W ) . We shall abbreviate the property of formal integrability by writing (2.1.1)

[ V, V] C V,

and refer to (M, V) as a formally integrable structure. An important subclass of the formally integrable subbundles are the integrable ones. D E fiN iT iO N 2.1.2. A subbundle V of C T M of dimension n is integrable if for any p e M there exist m = dimn* M — n smooth complex-valued functions Z 1, . . . , Z m defined in an open neighborhood Q C M of p with C-linearly inde­ pendent differentials d Z u . . . , d Z m such that LZ, = 0 for all L e f ( M, V) and 35

36

II. ABSTRACT AND EMBEDDED CR STRUCTURES

j = 1 , . . . , m. For po e M fixed, any such set of functions Zy, vanishing at po, will be called a family of basic solutions in Q. We note that if V is integrable, then (Af, V) is necessarily a formally inte­ grable structure. Indeed, if Li, L i e C°°(Af, W) , then [L\ %L 2] also annihi­ lates all the functions Zj given in Definition 2.1.2. By dimension, at any point p e £2 the space of complex tangent vectors annihilated by the complex covec­ tors d Z \ ( p ) , . . . , d Z m(p) coincides with Vp. Hence, the vector field [L1, L2] is a section of V. If the subbundle V is integrable, we shall refer to (A/, V) as an integrable structure on Af. For example, a nowhere vanishing complex vector field L on a manifold Af defines a formally integrable structure with fiber dimension 1. This structure is integrable if in a neighborhood of every point in Af there exist dim® Af —1 solutions of the equation Lh = 0 with linearly independent differentials. D EfiNrnoN 2.1.3. A formally integrable structure (Af, V) is called a formal CR structure if for all p e Af

(2.1.4)

y p n v p = {0 }.

We shall also refer to a formal CR structure as an abstract CR manifold, and to V as its CR bundle. A smooth section of V will be called a CR vector field on Af. A function (resp. distribution) on Af is a CR function (resp. CR distribution) if it is annihilated by all the CR vector fields on Af. The number n = dime Vp, for any p e M , is called the CR dimension of Af. If dimR Af = m -t- n as above, then the number d = m —n will be called the CR codimension of Af. l i d = 1, the CR structure is said to be of hypersurface type. The significance of the CR codimension is the following. If (Af, V) is integrable, then the basic solutions Z \ , . . . Z m provide a mapping from a neighborhood of po into Cm. This mapping is in fact an immersion (as is shown following Proposition 2.1.5), and hence the image manifold is of real dimension m -f- n and its real codimension in C m coincides with the CR codimension d = m — n. If a formal CR structure (Af, V) is integrable, we shall refer to it as an integrable CR structure or a locally embedded CR manifold. The following result connects the definitions of this section with those given in Chapter I. P r o p o sitio n 2.1.5. I f Af is a CR submanifold in C N and V is the CR bundle o fM defined in Chapter I, then (Af, V) defines an integrable CR structure.

P roof. By Definition 1.3.1, V is a subbundle o f C T M and it is easy to see that (Af, V) is a formal CR structure. We must show that it is integrable. Let po e Af and consider coordinates Z = (z, u/, w") e C N as in the proof of Theorem 1.8.1

§2.1. FORMALLY INTEGRABLE STRUCTURES ON MANIFOLDS

37

(see (1.8.3)). With the notation of (1.8.3), dime Vp = N — d\, for all p € M near po, and dim® Af = 2N — d. Hence, here we have m = N — d + d\. It follows from (1.8.3) that the restrictions ofthe coordinate f unct i ons ^, . . . , ZN-d,» w v • • • » w2d\-d have independent differentials near po = Oand, by definition, they are annihilated by any CR vector field on Af. This shows that V is integrable and completes the proof. □ The reader should note that, for an integrable structure (A/, V), the mapping £2 3 p h* Z(/?) € Cm, with Z( p) = ( Z i ( p ) , . .. , Z m(p)) where the Z} are a family of basic solutions in £2 C M as in Definition 2.1.2, is in general not of constant rank, and hence the image Z(£2) need not be a real submanifold of Cm, even if £2 is small around po. (Here, Cm is viewed as R 2m.) However, if we have an integrable CR structure, i.e. (2.1.4) holds, then the mapping /?h> Z{p) is of constant rank m + n and hence is an immersion. In this case Z(£2) is an embedded real submanifold of Cm of dimension m + n, provided Q is sufficiently small around pq. To see that p Z( p) is of constant rank m 4- n, we observe that (2.1.4) implies that dime Vp 0 Vp = 2n for every p 6 M. Moreover, we have (Vp © V p ) x = v ^ - n V p ,

and also that Vp and are respectively spanned by {id Z \ ( p ) , . . . , d Z m(p)} and {dZ\ (/?),... , d Z m(p)}. It follows that the dimension of the span of { dZi ( p) , . .. , d Z m(p), d Z \ { p ) , . .. , d Z m(p)} in C T * M is 2m — dim(V^- n ) = 2m — (m -f n — 2n) — m -F n. This observation justifies the interchangeable use of “locally embedded” CR manifold for “integrable” CR manifold. The following definition gives an important class of integrable structures which are not CR. DEfiNiTiON 2.1.6. A formally integrable structure (Af, V) is called real if fo r all p € M (2 . 1.1 )

Vp = Vp.

We have the following. T heorem 2.1.8. Any formally integrable real structure (Af, V) is integrable. P roof. We first claim that there is a subbundle S of T M , the real tangent bundle of Af, such that £ = C £ (= £C) with £ also formally integrable, i.e. [£, £] c £.

38

II. ABSTRACT AND EMBEDDED CR STRUCTURES

Indeed, if L \ , . . . , L„ is a basis of T(I/, V), where U is an open subset of M, then Re L j, Im L; , j = 1, . . . , n also span r ( U , V) over C by (2.1.7). Hence one can choose a local basis of real vector fields. The formal integrability of £ is an immediate consequence of that of V. The proof of the theorem now follows from the local Frobenius integrability theorem stated below. □ T heorem 2.1.9 (Local F robenius Integrability ). Let Q c R K be an open set and L \>. .. , L n smooth real vector fields in Q. I f L \ , . .. , L n are linearly independent at every p e Q and satisfy n

[L j, Lie] = ^ ^ CjkfLr, r=1

j, k = 1, . . . , n,

where cjkr are smooth functions in Q, then fo r any po e Q there exist smooth local coordinates x = (*i, . . . , x ^ ) near po such that r

V' '

^

i

r —l

where (ay>) is an invertible n x n matrix o f smooth functions near po. The proof of the above Frobenius Theorem can be found in many references and is based on elementary ordinary differential equation techniques. See, e.g. Chevalley [1] or Treves [2]. The following is an example of a structure which is neither real nor CR. E x a m p l e 2.1.10. Let M = R2 and L = —2 ix 2 g—. Note that L is nowhere vanishing and, if x = (*i, X2) € R2, then L x = L x if and only if x i = 0. The structure defined by L is integrable since we have L Z = 0 with Z(x) = x\ + ix \. The corresponding structure with n = 1 is often referred to as the Mizohata structure.

One may ask whether a formally integrable structure is necessarily integrable. This is true in the real-analytic category as is proved in Theorem 2.1.11 below. If M is a real-analytic manifold and the subbundle V of C T M is real-analytic, i.e. near every point po e M there exist real-analytic vector fields which form a local basis for T(M , V), and also formally integrable, then the structure (M, V) is said to be a real-analytic formally integrable structure. T heorem 2.1.11. I f (M , V) is a real-analytic formally integrable structure, then it is integrable. Moreover, near every po e M one can find real-analytic basic solutions Z \ , . . . , Z m (as in Definition 2.1.2).

We first state the Frobenius local integrability theorem for holomorphic struc­ tures, which is the analog of Theorem 2.1.9 in the smooth case and whose proof

§2.1. FORMALLY INTEGRABLE STRUCTURES ON MANIFOLDS

39

is based on similar arguments using ordinary differential equations techniques in the holomorphic setting. T heorem 2.1.12 (L ocal holomorphic F robenius integrability ) . Let £2, C C K be an open set and L i , . . . , L n holomorphic vector fields in £2 with holomorphic coefficients. l f L \ , . . . , L n, are linearly independent at every p e £2 and satisfy n

[Ly, Lk\ — ^ ^ CjkrL r,

j, k — 1, . . . , n t

r= 1

where cjkr are holomorphic functions in £2, then for any po G £2 there exist holomorphic local coordinates Z = ( Z i , ... , Z*) near p$ such that

where ( C T M / ( V ® V ) .

We observe that if X and Y are CR vector fields on Af, defined near p, with X (p) = X p € Vp and Y(p) = Yp e Vp, then np ([X, Y](p)) depends only on the values X p and Yp and not on the choice of the vector fields X and Y extending X p and Yp. (The proof of this is easily obtained by taking a local basis of the CR vector fields near p. The details are left to the reader.) D E fiN iT iO N

2.2.3. The Levi map at p e M is the Hermitian vector valued form Cp : V p x V p - > CTpM/ (Vp 0 Vp),

(2.2.4) CP( XP, Y P) : = i ^ ( [ X , Y](p)), where X and Y are CR vector fields on Af extending the CR vectors X p and Yp. Note that the definition above does not use the fact that V is formally integrable (involutive), see (2.1.1). However, we consider here only CR structures. Definition 2.2.3 yields a smooth Hermitian bundle map (2.2.5)

C: V x V

C T M / ( V © V).

We call C the Levi map of (Af, V). DEfiNiTiON 2.2.6. The Levi map Cp is nondegenerate if Cp(Xpi Yp) = 0 for all Yp e Vp implies X p = 0. If (Af, V) is of hypersurface type, i.e. dime Vp = (dim® Af — l)/2 , then the vector space on the right hand side of (2.2.1) is one dimensional. The Levi map at p is then a Hermitian form on Vp, called the Levi form. In this case, we say that (Af, V) is Levi nondegenerate at p if the Levi form is nondegenerate. Furthermore, a CR manifold of hypersurface type, (Af, V), is called pseudoconvex at po if the Levi form is either positive semidefinite at all p in an open neighborhood of po in Af or negative semidefinite at all p in such a neighborhood. Similarly, (Af, V) is said to be strictly pseudoconvex at po e Af if the Levi form is (positive or negative) definite at po € Af (and, hence, for all p e Af in a neighborhood of po). We shall discuss embedded pseudoconvex hypersurfaces further below. For a smooth generic submanifold Af c of CR dimension n and codimen­ sion d (see Definition 1.3.4), and a point po e Af, we shall describe the Levi map at po in regular coordinates Z = (z, w). for Af vanishing at po (as given by

II. ABSTRACT AND EMBEDDED CR STRUCTURES

42

Proposition 1.3.6). Thus, we assume that M is given near po = (0, 0) by (1.3.7). We parametrize M near the origin by (2.2.7)

R 2n x M d

b

(z, z, s) h* (z, 5 + *0 (z, z, s)) e M.

A basis for VPo is then

(2 .2.8)

d

d

dzi

dzn

and as a basis for € TpoM/ ( V Po 0 VPo) we can take (2.2.9)

dsi / ■ ”

os,

with the obvious identification. Given such bases, the Levi map can be represented by d Hermitian matrices of size n x n. P roposition 2.2.10. Let M C C N be a smooth generic submanifold o f CR dimension n, codimension d , and po e M. Let {z,w) e Cn x Cd be regular holomorphic coordinates fo r M vanishing at po so that M is defined near po = (0, 0) by (1.3.7). Then, the Levi map can be represented by the Hermitian matrices

( 2 .2 . 11)

(\ diZrj OvZ k' (0' 0' 0))'

t = 1, . .. ,d . \)) € C,

where, as in Definition 2.2.3, X and F are CR vector fields extending X p and Yp. Using the identity (see e.g. Helgason [1, Chapter 1, §2]) 2 (d), [X, P](p)) = X ((i9, ?)) |p - F ({0, X)) |p - 2{d0(p), Xp A Fp) = - 2 { f ( i d d p ) ( p ) , X p A Fp).

The last equality in (2.2.26) is a consequence of (2.2.25) and the identities (6 , X) = (0 , F) = 0,

46

II. ABSTRACT AND EMBEDDED CR STRUCTURES

which, in turn, follow since X, Y are CR vector fields and 6 is a real 1-form annihilating all CR vector fields and their complex conjugates. Writing a CR vector in the local coordinates Z, 0*3/ 9 Z*, we may identify Vp with the complex subspace V c of a 6 C N for which

(2.2.27)

The conclusion of Proposition 2.2.19 now follows from the definition of pseu­ doconvexity by writing the expression on the second line of (2.2.26) in the local □ coordinates Z. By Proposition 2.2.19, the conclusion of Theorem 2.2.17 is a direct consequence of the following proposition. P ro p o sitio n 2.2.28. Let M c C N be a smooth hypersurface with po e M. Let p ( Z , Z) be a local defining function for M near p 0 and Z) the distance to M as above. Then, there is an open neighborhood U C C N o f po such that (2.2.20) holds fo r all p e M fl U if and only if there is an open neighborhood U' c C N o f po such that —log 8m is plurisubharmonic in {Z e U ' : p(Z, Z) > 0}. Proof . It is straightforward to verify that if we replace the given defining function p (Z , Z) by p '(Z , Z) = c(Z, Z)p( Z, Z), where c (Z , Z) > 0 in an open neighborhood of po, then (2.2.20) holds for all p e M near p 0 if and only if the corresponding inequality with p replaced by p f holds for all p e M near po. We leave this calculation to the reader. Thus, we may replace p with one of the special defining functions o\ or 02 introduced in (2.2.14) (and the line following that equation). We choose o\ or o 2 according to whether the given defining function p is positive or negative in Q\. We shall assume, without loss of generality, that p is positive on £2i and, accordingly, we shall prove Proposition 2.2.28 with p replaced by a \ . For brevity, we shall write a — o\. Suppose that —log^A/ is plurisubharmonic in Pi \J \ for some open neigh­ borhood U 1 of po- We observe that —log 8m = —log o in . Moreover, a direct calculation shows that, for any a e C N,

(2.2.29)

§2.2. LEVI FORM AND LEVI MAP

47

Thus, we have in Q\ H U ' (2.2.30)

for every a e C N. We deduce that

(2.2.31)

holds for Z e Q \ D U ' and every a e C N with

(2.2.32)

Since a is smooth in a neighborhood of po with da ^ 0, we conclude, by passing to the limit in (2.2.31) and (2.2.32), that (2.2.20) holds, with p = a , for every p 6 M sufficiently close to po. This proves one implication in Proposition 2.2.28. To prove the opposite implication, we assume that the function —log 8m is not plurisubharmonic in Q\ D Uf for any neighborhood U' of po. Let U be an arbitrary open neighborhood of po- We shall prove that (2.2.20), with p = a , does not hold for all p € M fl U . In order to show this, we first construct an analytic disc (i.e. a holomorphic mapping of the unit disc in C into C N; see Chapter VII) which is contained in fl U except at one point p\ e M n U (interior to the disc) where the disc has quadratic contact with M. Using this disc, we then show that (2.2.20) fails to hold at p \ . By the assumption on —log 8m, there are points Z e U fl Q\, arbitrarily close to po, and X e C N such that the function of a single complex variable (2.2.33)

r

—log 8m (Z + X t, Z + X i)

is not subharmonic near r = 0. We choose Zq e Q\

(1 U, Xq e C N such that

By Taylor expanding the function in (2.2.33), we then obtain (2.2.34)

log^M(Z0 + tA.0, Z o + rA.0) = log 8m( Z0, Zo) + Re (ax) + Re (bx2) + e\ x \2 + 0 ( |r |3),

48

II. ABSTRACT AND EMBEDDED CR STRUCTURES

for some a, b e C. This can be rewritten (2.2.35)

&M(Z0 + tX0, Z0 + fX0) = SM(Z0, Z0) |ear+fct2| ^ l 2+0.

Since such Zo can be chosen arbitrarily close to po, we may assume that there is a point pi 6 U fl M which minimizes the distance from Zo to Af, i.e. (2.2.36)

Sm(Z o, Zo)

=

\ px — Zo|.

Consider the family of analytic discs A t : Ar -► C N, /4,(r) := Z0 + rX0 + (pi - Z Q)teax+bx\

(2.2.37)

where t e [0, 1], Ar = { r : |r | < r}, and r > 0 is a number to chosen. Observe that the distance to M has the following property SM(Z + ?, Z + f )
SM(Z0 + rA.0, Z0 + fX0) - /|p i - Z0| |eflr+&r2|.

By using (2.2.35), (2.2.36), and choosing r > 0 so small that e |r |2 + 0 ( | r | 3) > e | r | 2/2,

Vr e Ar .

we obtain (2.2.40)

SM ( A t( x ) , A M ) > SM(Z0, Z0)|eflr+fcr2| (e e|r|2/2 - t ) .

By shrinking r further if necessary, we deduce from (2.2.40), since A q (Ar) is contained in Pi (/, that A t (Ar) c fl U for all t e [0, 1). Thus, (2.2.40) implies that Ai(r) e Q\ C\ U, for all r e Ar \ {0}. By definition, we have Ai(0) = pi e M DU. We claim that

(2.2.41)

A d2o 2 2 ** Z 7 j dr *Z k'(P*’ PriaJa* > Q >

"

do

p f dZj

= 0*

where a = Aj(0). This would complete the proof. To prove the claim, let h ( r , r) = a ( A i ( r ) , Ai(r)). By the chain rule, (2.2.41) is equivalent to (2.2.42)

d2h —— (0, 0) > 0, d rdr

3h — (0, 0) = 0. dr

49

§2.3. CR MAPPINGS

Observe, since 8m and a coincide in constant C > 0 such that

that (2.2.40) implies the existence of a

h (r, f ) > C (ee|r|2/2 - t) .

(2.2.43)

By Taylor expanding at r = 0, we deduce that (2.2.44)

Re ( £ < 0 , 0 )e ) + I r . ( 0 ( 0 . 0 ) . ’) + ^ ( 0 . 0 , |r |’ > | | r |2 + e/2 > 0. Hence, (2.2.42) has been proved. This completes the proof of the claim and, hence, of the proposition. □ Although the focus of this text is, in some sense, on manifolds with degener­ ate Levi maps, we mention here without proof an important local embeddability theorem. 2.2.45 (K u r a n i s h i dim Af > 9, A k a h o r i dim Af = 7). Let (Af, V) be an abstract smooth CR manifold o f hypersurface type which is strictly pseudo­ convex. Then (Af, V) is integrable if dim Af > 7. T heorem

The proof of this deep theorem, which is outside the scope of this book, is due to Kuranshi [1] for the case dim Af > 9 and to Akahori [1] for the case dim Af = 7. We should mention here that the case dim Af = 5, with the notation of Theorem 2.2.45, is still open. When dim N = 3, the conclusion of Theorem 2.2.45 does not hold, as shown by a counter-example of Nirenberg [3], whose construction follows the lines of that of Example 2.1.15. §2.3. CR Mappings In this section, we shall discuss mappings between CR manifolds. We consider only those mappings that respect the additional structure induced on the manifolds by their CR bundles. As we will see, for embedded CR manifolds the typical example of such a mapping is the restriction of a holomorphic mapping of the ambient complex spaces. Let (Af, V) and (Af', V') be abstract CR manifolds with CR bundles V and V', respectively.

II. ABSTRACT AND EMBEDDED CR STRUCTURES

50

D E ftN iT iO N 2.3.1. A CR mapping (of class Ck, k > 1) H \ (M, V) is a Ck mapping H : Af —►Af' such that, for all p e Af,

(Af', V')

H*(Vp) c V ' Hip)i

(2.3.2)

where //* denotes the usual tangent map (pushforward) //*: TpM induced by H.

7//(P)Af'

It is immediate from the definition that if H : (Af, V)

(Af', V'),

G: (Af', V')

(Af", V")

are CR mappings, then the composition G o H is a CR mapping (Af, V) (Af", V"). The reader should observe that even if a CR mapping H : (Af, V) —►(Af', V') is a diffeomorphism as a C*-mapping Af Af', the inverse / / _1: Af' —> Af need not be a CR mapping (Af', V') -* (Af, V). Indeed, consider the following two CR manifolds: Af = C x R 2 with Vp at each point p being the complex tangent space of the first factor, and Af' = C 2 with Vp, at every p' being T®;1C 2. The identity mapping of the underlying space R4 is a CR mapping (Af, V) -> (Af', V'), but is not a CR mapping (Af', V') -> (Af, V). In this example the CR dimensions of the two manifolds are different. It is easy to see, however, that if a CR mapping H : (Af, V) —►(Af', V') is a diffeomorphism Af -> Af' and if the CR dimensions of (Af, V) and (Af', V') are equal, then the inverse mapping H ~ ] is a CR mapping. When the target manifold (Af', V') is embedded in C ^, i.e. Af' c is a CR submanifold and V' is the induced CR bundle as in Chapter I, then we have the following. P r o p o s it io n 2.3.3. Let (A f, V ) be an abstract CR manifold and A f' c C N a CR submanifold with induced CR bundle V '. I f H : Af -> A f' is a Ck-mapping, k > 1, with components H = (H \ , . . . , H^), i.e. Hj = Zj o H (for some choice o f coordinates Z in C N), then H is a CR mapping (A f, V ) -* (A f ', V ') if and only if each component Hj, j = 1, . . . , N, is a CR function.

We shall reduce the proof of Proposition 2.3.3 to that of a simpler case. Recall that the induced CR bundle V' of Af' C C N is defined as follows (2.3.4)

Vp, = C7>Af' 0 T°a C n ,

for each p ' e Af'. Hence, if a mapping H : Af -* Af' c is a CR mapping (Af, V) -* (Af', V'), then H considered as a mapping Af -» is a CR mapping (Af, V) (C ^, r a i C ^). Conversely, if a mapping / / : Af -* Af' C is a CR mapping (Af, V) -> (CN, 7 °^C ^), then it follows from (2.3.4) that it is also a CR mapping (Af, V) —►(Af', V'). Thus, Proposition 2.3.3 is a consequence of the following.

51

§2.3. CR MAPPINGS P r o p o s it io n

2.3.5. Let (Af, V) be an abstract CR manifold and H = ( HU ... , H n ) : M - + C n ,

N > 1,

a Ck-mapping, k > 1. Then H is a CR mapping ( M , V) —►(CN, 7'°-lC ;V) if and only if each component H j , j = 1, . . . , N, is a CR function. Proof.

Let L be a CR vector field on M. We have the following, for any

p e M,

j —\ \ dZJ

H {p)

H (p)/

Hence, H*{L{p)) e T ^ p)C N if and only if L //, (/?) = 0 for j = 1, . . . , N. It follows that H : M -> C N is a CR mapping (Af, V) (C N, T°•1M ) if and only if LHj = 0 for j = 1, . . . , TV and for each CR vector field L on M. This completes the proof of Proposition 2.3.5. □ When it is clear from the context what the CR bundles V and V' of M and M' are, we shall simply say that a mapping H : M -* Af' is CR if H : (M, V) -* (M', V') is a CR mapping. It follows from Proposition 2.3.3 and the basic material on CR functions in §1.4 that if M C C N and M f C C N> are CR submanifolds, H is a holomorphic mapping from an open subset U c C N into such that M C U, and H extends as a Ck mapping to M with H( M) C M \ then the restriction of H to M is a CR mapping from A/ into M'. We conclude this section with a result concerning embedded CR submanifolds. (A similar result can be formulated for abstract CR manifolds, but since we shall not need it, we treat here only embedded submanifolds.) Recall that if Af C C N is a CR submanifold, then the complex structure J of the ambient space restricts to a complex structure on Tj;M for every p e M. The following proposition states essentially that the tangent map induced by a CR mapping is complex linear on the complex tangent space. P r o p o s it io n 2.3.7. Let M c C^, Af' C C N> be CR submanifolds, and let H : M -* M ’ be a CR mapping. Then, fo r each p e M and X p e Tj;M,

(2.3.8)

( J ' o H * K X p) = ( H * o J ) { X p),

where J and J f denote the complex structure maps on T^M and T ^ p)M f induced from C N and C N', respectively.

52

II. ABSTRACT AND EMBEDDED CR STRUCTURES

P roof. Pick p e M and X p e Tf M. By Proposition 1.2.8, there exists L p € Vp such that X p = L p + L p. By linearity, we have (2.3.9)

H*(XP) = f t(L p ) + Hm( Lp) = H*(Lp) + H ^L p ).

Since HP(LP) e Vp, it follows, again by Proposition 1.2.8, that H*(XP) e Th(P)M'. Using the fact that Vp and Vp are the —i and + i eigenspaces for J (and similarly for J \ see Proposition 1.2.8), we compute (2.3.10)

H*(J(Lp + L p)) = H ^ - i L p + 1Lp) = - i H* { L p) + i H* { l p) = ~ i H*( Lp) + i lU L p ) = J' (H*(Lp)) + r ( H j L p ) ) = J' (H*(LP + L p)).

This proves Proposition 2.3.7.

□

Let us conclude this section with the following result, which is an easy conse­ quence of Proposition 2.3.7 (and Proposition 1.2.8) and whose proof is left to the reader. C orollary 2.3.11. Let M c C N, M f c C N be CR submanifolds and let H : M —> M f be a Ck mapping, k > 1. Then, H is a CR mapping if and only if (2.3.12)

C T cH{p)M \

V p e M.

§2.4. Approximation Theorem for Continuous Solutions Let (Af, V) be an integrable structure as given in Definition 2.1.2. In this section we shall study the solutions of this structure, i.e. the functions or distributions on M which are annihilated by all the vector fields L 6 T(Af, V). Note that by the chain rule, any holomorphic function of the basic solutions Z x, . . . , Z m (given in Definition 2.1.2) is in particular locally such a solution. Conversely, the main result of this section shows that in fact all solutions can be locally approximated by holomorphic functions of the basic solutions Z; . Theorem 2.4.1. Let (Af, V) be an integrable structure, po e M, and Z = (Z i , . . . , Z m) a family o f basic solutions near Pq. Then there exists a compact neighborhood K o f po in M such that fo r any continuous solution h in M, there is a sequence o f holomorphic polynomials Pv(z) in m complex variables with the property that (2.4.2)

h(u) = lim Pv( Z( u)), V —* 0 0

u e K,

§2.4. APPROXIMATION THEOREM FOR CONTINUOUS SOLUTIONS

53

where the convergence is uniform in K. If h is a solution of class Ck, k > 1, then the convergence in (2.4.2) is in Ck( K ) as will be shown in §2.5. Proof of T heorem 2.4.1. We begin by making an appropriate choice of local coordinates near the central point po. L emma 2.4.3. Let (M, V), po> and Z = ( Z i , . .. , Z m) as in Theorem 2.4.1. Then, after an invertible complex linear transformation in the Zj one can find local coordinates (x, y) near po, vanishing at po, with x = ( x \ , ... , xm) and y = , . . . , y n ) t such that (2.4.4)

Z j ( x t y) = xj 4- ii, . . . , vm) e Cm, we use the notation v2 = v • v = vj. We fix y e R n with 0 < \y\ < d and let D = Dy be the (m -j- l)-dimensional manifold D := {(*', y ') € R k : \x'\ < r, y' = ty, t e (0, 1)}.

(2.4.6)

By Stokes’ Theorem we can write

f

(2.4.7)

d a v(x', / ; z) = f

JD

a v( x\ / ; z).

J dD

We shall first compute the right hand side of (2.4.7). Since x ( x ') = 0 for \x'\ > r we have (2.4.8)

f

« ( * ', / )

J dD

/ y \ m/2 /* = (-) / exp[-v(z - Z(x', >-))2] x ( x ' ) h( x\ y)dX’Z(x' , y) / y \ m/2 Z4

- ( - ) VX '

/

exp[—y(z — Z(x', 0)) ] x(x' )h(x' , 0)dx>Z(x', 0),

J R'«

where means taking the differential with respect to the x r variables. In order to compute the left hand side of (2.4.7) we need the following lemma. L e m m a 2.4.9. Let £2 be an open subset o f R kf k = m + n, and (£2, V) an integrable structure, with basic solutions Z \ , . . . , Z m defined in £2. A distribution f in £2 is a solution if and only if d ( f d Z ) = 0 in £2, with d Z ( u ) = dZ\ {u) A . . .

A

d Z ffi (m).

P r o o f . Assume first that / is of class C 1 in £2. Then / is a solution if and only if, for every p e £2, d f { p ) e Vp. The latter is equivalent to the statement that d f ( p ) is in the span of {d Z \ ( p ) , . . . , d Z m(p)}. Since d ( f d Z ) = d f a d Z , the lemma follows in the case where/ is of class C l . By shrinking£2 if necessary and using Lemma 2.4.3 we may assume that the Z /s are given by (2.4.4). Then d Z \ , . . . , d Z nud y \ , ... , d y n form a basis of C r (* v)£2 at every point (x, y) e £2. Hence if follows that there exist vector fields Rj, 1 < j < m and Sq, 1 < q < n, such that for any / of class C 1 we may write

m

(2.4.10)

df =

n

Rj f d Z j + J 2 s q f d y q. y=l

9=1

From the above it follows that / is a solution if and only if 5^ / = 0, q = 1, . . . , n, and hence the vector fields Sq form a basis of the sections of V in £2. For a

§2.4. APPROXIMATION THEOREM FOR CONTINUOUS SOLUTIONS

55

distribution / in £2, (2.4.10) still holds, and hence / is a solution if and only if d f is a linear combination of the dZ j with distribution coefficients. The rest of the argument for a general distribution is the same as in the C 1 case. This proves Lemma 2.4.9. □ We return to the calculation of the left hand side of (2.4.7). We observe that since exp[—v(z — Z { x \ y'))2] is a holomorphic function of the Zy(jt\ y') and that the product of two solutions is again a solution, we conclude by Lemma 2.4.9 that

(

y\

m/2

-J

ex p [-v (z - Z (x ’, y '))2] h(x', y' )dx(x' ) A dZ(x' , y').

We introduce the sequence of entire functions in Cm defined by (2.4.11)

Hv{z):= / y \ m/2

f

(-)

/

exp[—v(z — Z(x' , 0)) ] x{x' )h{x' , 0)dX' Z( x' ,0).

We shall show that Hv( Z ( x , y)) converges uniformly to h(x, y) on K as v oo. For this, we shall prove, after replacing z by Z ( j c , y), that the left hand side of (2.4.7) converges uniformly to 0 on K as v -> oo, while the first integral on the right hand side of (2.4.8) converges uniformly to h(x, y). This will be proved in the following two lemmas. L e m m a 2.4.12. I f r and d are chosen to be sufficiently small, then fo r any continuous function f in Q,

(2.4.13)

f

ex p [-v (Z (x , y) - Z (x ',y '))2] f (. x' , y' )dx(x' ) A d Z ( x ' , y ' )

0

Jd

uniformly fo r (jc, y) € K as v -* oo. Note that since d x ( x ) = 0 for \x\ < j and |j c | > r, the integral in (2.4.13) is evaluated over the set £ < \x'\ < r and y' € [0, y]. The main point is to estimate | exp[—v(Z(jc, y) - Z(jc', y'))2]|- Using (2.4.4) we have P ro o f.

(2.4.14)

Re (Z(x, y) - Z(x' , y'))2 — (x - x ')2 - ((x, y) - (x\ y '))2.

By the mean value theorem we have \ 3 //3 s(0 , t) satisfies the ordinary differential system

(3.1.29)

du A 3Y — = X ( f { 0, 0 ) 4- V — ( /(0 , 0)«y, dt 9xj

«(0) = 0.

The proof will be completed, in view of the uniqueness of solutions of differential equations, by showing that the function t t X ( f ( 0, /)) also solves (3.1.29). Clearly, this function satisfies the initial condition. We differentiate it and obtain (3.1.30)

L (tX ( f ( 0 , /))) = X ( f ( 0, t)) + t T dt “

f ^ ( / ( 0, t))-:fM 0 , t). dxj at

§3.1. NAGANO’S THEOREM

67

Using (3.1.26) and the definition of the commutator, we obtain Vp € W \

In view of Lemma 3.1.14, [7, X ](/(0 , t)) = 0 for all r, and hence it follows from (3.1.31) that tX ( /( 0 , t)) satisfies (3.1.29). Claim 3.1.24 follows from uniqueness □ as explained above. Since f ( s , t) maps the rectangle R into W, [df/ds](s, t) is in the tangent space Tf{Sj)(W). In particular, it follows from (3.1.25) that X( p) is in Tp(W). Since p e W and X e y were chosen arbitrarily, the proof of (3.1.5) is complete in view of (3.1.21). It remains to prove the uniqueness part of Theorem 3.1.4. Suppose that W f is another real-analytic manifold in £2 through p0 satisfying (3.1.5). Necessarily, dim W f = dim IT = dimg(O). Hence, it suffices to show that there is an open neighborhood V of 0 in £1 such that

(3.1.32)

wn v c w'n v.

Let U be a convex open neighborhood of 0 in U c Mr (where the map (3.1.20) is defined) and define W C W to be the manifold defined as the image under (3.1.20) of U . We choose an open neighborhood V of W in Q such that W H V = W. We can choose U and V so small that W' C1 V is closed in V. In order to prove (3.1.32), we pick p\ e W. By the construction of W, there exists Y e g such that the integral curve y(t) = exp0(/7 ) goes through p\ at time one. Note that y(t) € W C V for t e [0, 1] since U was chosen to be convex. Since (3.1.33)

Y(p) e Tp(W' )f

by assumption, it follows that y(t) e W f fl V for t sufficiently small. We need to prove that y ( 1) € W' fl V. Let iE = {t0 e [0, 1]: y(t) e Wf fl V , Vf e [0, f0]}. The set E is open by (3.1.33). It is easy to check that E is also closed since W' fl V is closed in V. The inclusion (3.1.32) is proved, and the proof of Theorem 3.1.4 is complete. □ 3.1.34. If, in addition to the assumptions of Theorem 3.1.4, one as­ sumes that dim g(p) = dim g(p0) for all p e Q, then the conclusion of the theorem is a consequence of the Frobenius Theorem (Theorem 2.1.9) in the real-analytic setting. R em ark

68

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

§3.2. Sussman’s Theorem In this section we prove a theorem of Sussman, which may be viewed as an analog of Nagano’s Theorem for sets of smooth real vector fields. It may also be regarded as a generalization of the Frobenius Theorem (see Remark 3.2.24 below). Let V be a collection of smooth real vector fields in an open set £2 C R n. By a polygonal path o f J integral curves of vector fields in V joining p e £2 to q € £2, we mean a piecewise smooth curve y : [0, 1] £2 with y (0) = p, y( 1) = q, and 0 = sq < s\ < . .. < sj = 1 such that, for j = 1, . . . , J , y(s) = expY(s. 0 t(s)Xj, S j - i < s < S j , where Xj e V and t (s) is a smooth diffeomorphism of [sj- \ , Sj] onto some closed interval of R with fCsy_i) = 0. If the collection V consists of real-analytic vector fields then we require the curve y to be piecewise real-analytic. T h e o r e m 3.2.1 (L o c a l S u s s m a n T h e o r e m ). Let £2 be an open set in R n and let po e £2. Let V be a collection of C°° vector fields in SI. Then there exists a C°° submanifold W o f SI with po e W such that every vector field in V is tangent to W at all points o f W and such that the following hold:

(i) I f W f is a C°° submanifold o f £2 containing po and to which all vector fields in V are tangent at every point o f W f, then there exists an open neighborhood V C £2 o f po such that W n V C W' fl V. (ii) For every open set U C SI containing pq, there exists a positive integer J and open neighborhoods V\ C Vi C U o f po such that every point p e W fl V\ can be reached from po by a polygonal path o f J integral curves o f vector fields in V contained in W fl Moreover, if the vector fields in V are real-analytic, then W is also real-analytic. R e m a r k . Since, by (i) of the theorem, the germ of the manifold W at po is unique, we shall refer to it as the local Sussman orbit of po (relative to V). Note that the local Sussman orbit does not change if we replace SI by any other open subset £2' C SI containing p0.

Before we enter the proof of Theorem 3.2.1, we need to introduce some notation. We shall use the notation £ = (X \ , . .. , X k) e V k and T = (t\ , . .. , tk) e R*, where k is a positive integer that will vary throughout this section. We denote by (T , p) (/?), the mapping R k x £2 -> £2, defined for T near 0 in R*, by (3.2.2)

4>£>r(p) := exptkX k • expt*-i**_i • . . . • expt xX x • p,

where we use the notation (3.2.3)

exp t X • p := exp^ tX.

69

§3.2. SUSSMAN’S THEOREM

Now, let U be an open subset of £2. For fixed p e U and £ as above, the mapping T h* we obtain an open subset A^ c R* x U in which the map (3.2.2) is defined and maps into U. For U c £2 as above, we denote by Gx>(U) the pseudogroup of local diffeomorphisms generated by the vector fields in V. More precisely, Gv ( U) consists of all the mappings (3.2.6)

p h* &$,T(P)

for all possible choices of a positive integer k, £ e V k and T € U peU p c R*. Note that the mapping (3.2.6) is not necessarily defined for all p € U. We take the domain of the mapping (3.2.6) to be precisely those p e U for which T is in A ^p. This means that we require not only that the image of p under (3.2.6) is in U but that the entire polygonal path of integral curves joining p to ^ ^ .r(/?) stays in U . The mapping (3.2.6) is a diffeomorphism from its domain onto its image. The inverse of the mapping is given by (3.2.7) where (3.2.8)

( ^ . r ) _1(^) = exp—t \X\ • exp—t i Xi • . .. • exp—tkXk • q , is given by (3.2.2). We shall use the notation I : = ( X k, X k- U . .. , * i ) ,

T := (tkj k. u . .. , r0 ,

where £ and T are as in (3.2.2). It follows from (3.2.7) that ( ^ . r )” 1 = JrFor p e £2, we denote by A (p) c TP(Q) the linear span of V{p), i.e. the tangent vectors at p obtained by evaluating the vector fields of V at p. These subspaces A (p) define a C°° distribution in £2, although we shall not make use of this fact here. Now, let U be an open subset of £2 as above. We shall define another

70

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

collection of linear subspaces of Tp (U) which will depend on V and U . For p € U, let V%(p) be the smallest Gx>(U)-invariant subspace of TP(U) containing A (/?), i.e. 'P p(p) is the linear span of the union of all (3.2.9)

( ^ .r ) * ( A ( 9 ))f

with £ 6 P*, T e (Jgei/ Q *n domain of as described above, and 4>£.r(, then dim 'Pp(p) = dim V!p(q), since the inverse of $,7 e Gp( G) is also in G p ((/). For po in the domain of ^ 70, and go = 4>s.70(Po)> we denote by VpolSo C Tqo(U) the subspace (3.2.10)

V e .r„ :=0»(7yo(K*)),

with 0( T) = o(po)) e r o(p0)), e Gp((7), and 4>^/ r o(p0) is in the domain of 0, which together imply that 0*O/3f/) € Vg(qo). □

§3.2. SUSSMAN’S THEOREM

71

Lemma 3.2.16. There exists a positive integer k, a vector £ e V k, and 7 ° e R k with po in the domain 0 € Gx>(U) such that *W .ro = Pg(po). P r o o f . We start by choosing a positive integer ki, £ ' e V k\ and T,° 6 R * 1 such that po is in the domain of 1,7-0 £ Gv ( U) and such that = po (we can take e.g. any k,, any £ ' e V k\ and r,° = 0 e R*1)- Now, in view of Lemma 3.2.11, either Vpo^1 To equals V^(po) or there is a tangent vector YPo e

K ( P o ) \ vPor . To. In the first case, the proof is complete. Thus, we assume that there is YPo e P^ i po) \ the construction of V%(po)> there is £2 € V kl, 72° g R*2, a point # in the domain of 4>^2 To e Gx>(I/), and Yq e A(q) such that YPo = (£,:r0(7o) = 7o- We claim that VPo^.t° contains the linear span of Vpi j:i yO and Ypo- To see this, we differentiate the map 7 h> 4>^y(/?o), denoted by 0 (0 as in (3.2.10), with respect to T\ and t at 7°. As in the proof of Lemma 3.2.11, we obtain, in view of (3.2.17), (3.2.18)

e .(

) = ( * ? .,* ) ,Y(q) =

and it is easily verified from the definition of £ and 7° that

;=i and hence, yq (po) = 0. We complete the proof by induction on i for the following property. (*)£ For any commutator Z o f any length i as in Lemma 3.4.22, yy(w), defined by (3.4.23), vanishes to order at least i f o r all j such that mj — i > I. We have already proved that (*)* holds for i = 1. Clearly, Lemma 3.4.22 follows for a given commutator Z of length i and a given j from (*)^ with t = m7 - i. Assume (*)* holds for i = 1, . . . , £0 for €0 < tnh — 1 (for £0 > nth the condition (*)*0 is vacuous for all commutators). We shall prove it for i = to -f 1. For this we fix a commutator Z of length i and assume that (3.4.23) holds. We must show that, for any string P = X Vl . . . X VfQ of length i 0 and any j 0 such that (3.4.26)

W/0 - i > fo + 1,

we have (3.4.27)

(^y ;0)(po) = o.

80

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

We fix a string P of length to and jo as above. Consider the commutator of length i + io (3.4.28)

/ ? : = [ X Vl)[Xy2. . . . [ X lVo,Z ]...]],

and decompose it as h

(3.4.29)

R = a(u) • X +

y, (u) • Sj. j =i

Observe that, since ntj0 — i > to -f 1 implies m7o — (A) + 0 >: 1. condition (*)i implies (3.4.30)

Yj0(0) = 0.

On the other hand, using the decomposition (3.4.23) of Z and expanding the right hand side of (3.4.28) we obtain h

(3.4.31)

R = (P a ) ( u ) ■X + £ ( P y 7)(«) • Sj + . .. , j =1

where the dots . .. denote a finite sum of terms of one of the following two forms (3.4.32)

(P ,« )(« )• [X '.X * ],

( P > p)( u ) - [ X , l >Skp\,

with Pq, Pq strings of length q < to»and X 7, X'7 commutators of length to —q of vector fields in V. We decompose such terms as follows (Pqa) ( u) [X' , X p] = (Pqa ) ( u ) ( a Hu ) X + J ^ y q(u) ■s j j ;= ‘ h

(3.4.33)

(P'YkPm [ X ' 1, Skp] = {P'qykp)(u) ■( a ’q(u) ■X + £ > 'J (u ) • S,).

To prove (3.4.27), in view of (3.4.30) and (3.4.31), we must show that (3.4.34)

(Pqct)(po)Y?0(Po) = 0,

(P ',y ^ )(p o )y 7 0(po) = 0.

First, since [X7, XM] is a commutator of length to —q + 1 and my0 —(to —g -f-1) > i + q > 1 by (3.4.26), it follows from condition (*)i that y/0(po) = 0. To prove the second identity in (3.4.34), which will complete the induction, we recall that y*p(u) is the coefficient of SkP in the decomposition of the commutator Z, which is of

§3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINITE TYPE

81

length i . By condition (*)?+1, which is assumed to hold by the induction hypothesis since q < to, it follows that (PqYkp)(po) = 0, provided that mk —i > q + 1. On the other hand, if m k —i < q + 1, which is equivalent t omk —q < i , then it follows from (3.4.26) that ntjQ- (jmk + to - q) > / + to + 1 - (mk - q + to) > 1. In this case, since y'J0(w) is the coefficient of S;o in the decomposition of [ X fI, S*p], which is a commutator of length mk + to —q, we have y'J0(po) = 0 in view of the condition (*)i. We have proved that condition (*)£0+i holds. This completes the proof of Lemma 3.4.22. □ If D (x , s, t) is a formal power series in (x , s , t) with coefficients that are dif­ ferential operators in Q, then we shall write D (x , s , t) = 0( k ) if all monomials in this power series are weighted homogeneous, in (;t, s , t), of degrees at least k. The next step in the proof of Theorem 3.4.7 is the following lemma. 3.4.35. Let D ( j c , s , t) be a formal power series in ( j c , s , t), with no constant term, with coefficients that are vector fields in Q. Write L em m a

mh (3.4.36) D (x , 5, f) = P (x , j, 0 + 0(w/, + 1) = ^ 2 pk(x, s, t) + 0 { m h + 1),

k=\ where Pk{x, s , t) is a weighted homogeneous polynomial o f degree k with coeffi­ cients that are vector fields in Q. For each k, 1 < k < mk, write (3.4.37)

Pk(x , s, t) = y ^ q [ ( x , s, t)af(u)Xj + ^ p kJp(x, s, t)yfp(u)Sjp. i )P

Denote by #*(D) the number o f terms in (3.4.37) fo r which the corresponding coefficient cef(u) or Yjkp(u) is nonconstant in Assume that, fo r every k = 1, . . . , m h, (3.4.38)

k + o(yjkp) > m j ,

V j = l,...,h , V p — l,...,£j.

Also, assume that fo r some I q > 1, (3.4.39)

V i < £o-

# ,( /» = 0,

If #i 0(D) 7^ 0 let r(x, 5, t)fi(u)Z be a vector field chosen among { (< ? ;„ (* .t)aj°(u)Xi)r

(p]tP a (x ,5, t)yj°(u)Sj,p)j p}

such that 0(u) is nonconstant in £2. Then (3.4.40)

Exp(D(x, j , t)) = Exp(r(x, s, t)(p(u) - /i(po))Z) Exp(D’(x, s , t )),

82

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

w h e r e D \ jc, s , t ) h a s th e s a m e p r o p e r tie s a s D ( x , s , t), i.e. s a tis fie s th e a n a la g o u s a s s u m p tio n s ( 3 .4 .3 6 - 3 .4 .3 9 ) . I n a d d itio n , w e h a v e (3 .4 .4 1 )

#e0( D ,) = # €o( Z » - l

a n d D ( x , s , t ) — D ' ( x , 5, / ) = O (^ o )P r o o f . W e s h a ll p r o v e th e le m m a o n ly in th e c a s e w h e re r ( j t , s , t ) f i ( u ) Z = p { l ( x , s , t ) ( y - p ( u ) ) S j p f o r s o m e y , p , a n d le a v e th e o th e r c a s e to th e re a d e r. A p p ly in g th e B a k e r - C a m p b e l l- H a u s d o r f f f o r m u la in T h e o r e m 3 .4 .9 , w e o b ta in ( 3 .4 .4 0 )

in w h ic h

( 3 .4 .4 2 )

D '(x ,s,t)

= P ( x , s, t ) - p { p ( x , s, t ) ( y l p ( u )

- Y j j } ( Po ) ) S Jp + C ( x , s , t ) + 0 ( m h + 1),

w h e r e C ( x , s, t ) is a fin ite s u m o f c o m m u ta to r s o f

P io i x ' s
))Sjp

-

w ith th e r e m a in in g te r m s in P ( x , s , t ). It is e a s y to se e th a t C ( x , s , t ) is 0 ( € o - f 1), w h ic h p r o v e s th a t ZX(jc, s , t ) — D ( x , s , t ) = 0 ( i o). H e n c e , s in c e w e h a v e re m o v e d o n e n o n c o n s ta n t te rm in Pe0( x ,

r), it is c le a r th a t th e p r o o f o f th e le m m a re d u c e s

to c h e c k in g th a t e a c h c o m m u ta to r in C ( jc, s , 0 is d e c o m p o s e d a s ( 3 .4 .4 3 )

< ? * (* -t)a-(u)Xj + ^

D

p [ p { x , s, t ) y f p (.u)Sj p ,

jP

*

f o r s o m e k > t o - f 1, w h e re e a c h q lk ( x , s , t ) a n d p kJ p ( x , s, t ) a re w e ig h te d h o m o ­ g e n e o u s p o ly n o m ia ls o f d e g r e e k , a n d s u c h th a t k + o(Yjkp ) > m j ,

( 3 .4 .4 4 )

f o r a ll j = 1 , . . . , h a n d p = 1 , . . . , l j . W e c o m p le te th e p r o o f a s fo llo w s . L e t p k ( x , s , t ) a ( u ) Y a n d q t ( x ,

5, t ) f i ( u ) Z ,

w h e re T, Z € { (X ,) , ( 5 ^ ) } , s a tis fy ( 3 .4 .4 5 )

^ + o (a ) > m ( f ) ,

/ + o (/J ) > m ( Z ) ,

w h e re /? (* , s , r) a n d g ( j t , s , r) a re w e ig h te d h o m o g e n e o u s p o ly n o m ia ls o f d e g r e e k a n d I , re s p e c tiv e ly , a n d 1 ( 3 .4 .4 6 )

m ( U ) :=

if U = X , fo r so m e i = 1 , . . . , r, i f U = Syp f o r s o m e j = 1 , . . . , A, and p = 1 , . . . , f y.

§3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINITE TYPE

83

Expansion of their commutator gives (3.4.47)

[pk(x, s, t )a(u) Y, qt (x, s, t)P(u)Z]

= p k(x, s, t)qi(x, s, t ) (a(u)(Y/3)(u)Z - P(u)(Za)(u)Y + a(u)P(u)[Y, Z ] ) . Decomposing the commutator [F, Z] according to (3.4.23), we obtain (3.4.48)

[pkaY, qifiZ] = p kqt ( a ( Y p ) Z - P( Z a ) Y + a/}(8 • X + £

€j ■5 ,)) ,

where 8 = 8(u) and €j = tj(u). By Lemma 3.4.22, we have (3.4.49)

o(€j) > rrij - (m(F) + m(Z)).

We claim that each term on the right side of (3.4.48) can be written in the form p'k,(jc, s , t )a' (u)Y', for some F' € {(X,), (S/p)}, where (3.4.50)

k' 4- o(or') > m (F').

We leave it to the reader to verify this for the terms pk qici{YP) Z, pk qt P (Z a ) F, pk qe a p 8 • X, and we shall check it for the term p'k, a' • F' = pk qe a p €j • . In this case, we have (3.4.51)

*' + o(of') > * + 1 4- o(a) + o(j8) + o(€j) > m(F) 4- m(Z) -f m;- — (m(F) 4- m(Z)) = m7- = m (F'),

where in the second line we have used (3.4.45) and (3.4.49). This completes the proof of the claim. Lemma 3.4.35 follows from the claim by induction. □ In order to use Lemma 3.4.35 in the proof of Theorem 3.4.7, we need the following observation. L e m m a 3.4.52. Let D(x, s, t) be a formal power series in ( x , s , t ) with coef­ ficients that are differential operators in £2, and let Z be a vector field in £2 such that Z(po) = 0. Then the following identity holds, as formal power series in (x , s , t, ?),

(3.4.53)

^Exp(CZ) Exp(D(x, s, 0 ) / ) ( p 0) ~ ^Exp(D(x, 5, r))/)(p o ).

84

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

fo r every smooth function / ( « ) in £2. P r o o f . B y e x p a n d in g th e fo rm a l p o w e r s e r ie s , w e o b ta in

(3.4.54) Exp«Z )E xp(D ') , o,

where h( 0 , w) is the holomorphic function obtained from h( 0 , w) by replacing the coefficients in the Taylor expansion by their complex conjugates. To solve (4.2.9), we seek a solution h(z, w) satisfying also h(0, w) = - h { 0, w)y i.e. the coefficients of h(0, u;) are pure imaginary. For this, we substitute for h(0, w) in (4.2.9) to obtain (4.2.10) h(z, vo) - ( ~ h (0, w)) - 2i ^z, 0, w + — ■— ■

— ^ s 0.

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

98

Putting z = 0 in (4.2.10) we obtain h(0, w) = i'0(0,0, w). Substituting this for h(0, w) in (4.2.10) we obtain (4.2.11)

h(z, w) - (-iT) defined in a neighborhood o f 0 in C2n+d>satisfying Q (z, 0, r) = respectively. After a linear change of coordinates in C ^, we may assume that the components of c are of the form X(z, z, s, s') = z + 0(2 ) (4.3.7)

rk( z , z , s , s ' ) = sk + f k( z , z , s u . . .

+ 0 (m k)

r \ z , z, s, s') = s' + / '( z . z. *) + 0( m) , where f k(z, Z, -Vi, . . . , j*_i) is a polynomial, without linear terms, which is a sum of weighted homogeneous polynomials of degrees between 2 and m k — 1, for k = 1, . . . , h. Similarly, f ' ( z , z, s) is a polynomial, without linear terms, which is the sum of weighted homogeneous polynomials of degrees between 2 and m —1. The remainders 0(2), 0 ( m k), and 0 ( m ) do not have any linear terms. In what follows, we shall use the notation 0 ( v ) for a remainder that is 0(v), v > 2 , and that, in addition, does not have any linear terms. Of course, if v > m, then a term which is of weighted degree v cannot have any linear terms and, hence, the two notions O(v) and 0 ( v ) coincide. It is sometimes convenient to denote r ' by z>,+i and m by mh+ \. The following lemma will allow us to eliminate the polynomials f k and / ' in (4.3.7).

104

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

Lemma 4.3.8. Assume, for some k, 1 < k < h -j- 1, that a local embedding i = (x, n , ... , T/j+i) o/A/ into C N is given of the form x ( z , z , s , s ' ) = z -F 0(2) xj (z, z, s, s' ) = Sj + ifmj( z , z , s u . . . ,Sj-x) + 0( mj + 1),

(4.3.9)

j = \,...,k-l xk(z, z, s, s') = sk + rd ( z , z , s u ■■■, s k- 1 ) + 0( f i + 1) xt (z, z, s, s ’) = st + 0(2),

i

= k + 1,... ,h + 1,

where \//mj(z, z, s i , ... , sy_i) is a vector valued weighted homogeneous polyno­ mial ofdegree m}, rM(z, z, s Xl... , s*_i) is a vector valued weighted homogeneous polynomial, without linear terms, 0/ degree p, and p < mk. Then there is a weighted homogeneous polynomial P( x, t x, ... , r*_i), without linear terms, of degree p such that, after the following polynomial change of coordinates in C N,

(4.3.10)

*

rk = rk - P( x, t \ , ... , rk-\),

, S J S * + '•

the imbedding 1, in the new coordinates (x, xj, ... , r/*+i), is of the form (4.3.9) with the corresponding = 0. P r o o f . First, we claim that the vector fields L y. _ i , for j = 1,... , n, commute with each other. Indeed, since the Ly are assumed to commute, we have

(4.3.11)

0 = [Lj, Lt\ = [Lj - \ , L i - \ ] + 0 ( - 1).

By identifying the terms of weighted degree —2, the claim follows. Since the components of the embedding 1 satisfy the system Lju = 0, j = 1,... ,n, it follows, by taking the lowest weighted homogeneous term in Lj i i = 0 for t = 1,... , k, that for j = 1,... , n L j - \ (se -F ^mt(z, z , s u . . . , ^ - 1 )) = 0

,

£ = 1,

1

Ly._irM(z,z, ji, ... ,$*-1 ) =0. We introduce the truncated vector fields M y _i obtained by replacing the sum in the right side of (4.3.5) with a sum of only the first k — 1 terms, i.e.

(4.3.12)

9

Mj,-i = — + E ^ ( z>2>-si> • • • dZj

9

■—

,

j = l , . . . , n.

105

§4.3. CANONICAL COORDINATES FOR GENERIC SUBMANIFOLDS

Since the L; ._i commute with each other, it is clear that the M/,_i also commute. Observe that, for any / , M y- - i / ( z , z, $ i , .. .s*-i) = L/ ,_i / (z, z, $ i , .. .s*~i)and, hence, + l / f mei z , Z , S \ , . . . , 5 * - i ) ) = 0, JL f ,( z , -z , 5 i , . .. , sk- 1)A=H0. My,_irM

(4*3-13)

i

=

1, . . . k

-

1,

Note that the coordinate functions Zi , . . . , zn are also annihilated by the vector fields M j - 1. It follows that the components of the complex vector-valued func­ tions z, st 4- \/fmt (z, z , s \ , . .. , ^_i ) , i = 1, . . . , k — 1, form a basis of linearly independent solutions of the system M j - \ u = 0, j = 1, . . . , n (in the restricted variables z, z, t = 1, . . . , k — 1). We replace z by n independent complex variables and consider the My,_i as holomorphic vector fields C2n+k~l . Since the vector fields M j-1 are linearly independent and commute with each other, and since rM(z, z, s\ , ... , sk-\ ) is a polynomial solution of the system M j - \ u = 0, j = 1, . . . , n, we may apply the holomorphic Frobenius theorem (Theorem 2.1.12) to conclude that there is a holomorphic function P ( x * ri> • • • >T* -i) such that (4.3.14)

r/x (z,

z, s i , . . .

,

s*-i)

= P(Z,Sl + ^mife.Z). ••• .**-1 4-^«*_1( z , z , J i , . . . , ^ - i ) ) . Using the weighted homogeneity of and the fact that r^ has no linear terms, one can easily check that P is a weighted homogeneous polynomial without linear terms. With this choice of P( x , Ti , ... , r*_i) in (4.3.10), the reader can verify that the conclusion of the lemma follows. □ By repeatedly using Lemma 4.3.8, we may assume (4.3.7) holds with f k == 0 and / ' == 0. We write X ( z , z , s , s f) = z 4- 0(2)

(4.3.15)

rk(zt z , s , s ' ) = sk + \/rmk(z, z , $ i , . .. ,s* -i) 4-