Differential Equations on Complex Manifolds

516 Pages·1994·21.286 MB·English

by  Boris Sternin, Victor Shatalov (auth.)

Differential Equations on Complex Manifolds
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 276
Differential Equations
on Complex Manifolds
by
Boris Stemin
Moscow State University,
Moscow, Russia
and
Victor Shatalov
Moscow Institute of Electronics and Mathematics,
Moscow, Russia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4368-9 ISBN 978-94-017-1259-0 (eBook)
DOI 10.1007/978-94-017-1259-0
Printed on acid-free paper
All Rights Reserved
© 1994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1994
Softcover reprint of the hardcover l st edition 1994
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface xi
Introduction 1
0.1 Generalities .................... . 1
0.2 The Integral Transformation ........... . 4
0.3 Differential Equations with Constant Coefficients 14
0.4 Elementary Solutions .............. . 22
0.5 Continuation of Solutions to Elliptic Equations 30
0.5.1 Statement of the Problem ....... . 30
0.5.2 Complexification of the Continuation Problem 36
0.5.3 Examples . . . . . . . . . . . . . . . . . . . . 38
1 Some Questions of Analysis and Geometry of Complex
Manifolds 41
1.1 Summary of Results 41
1.1.1 Leray's Residues 42
1.1.2 Function Spaces . 46
1.1.3 Parameter-Dependent Integrals 49
1.1.4 Homogeneous Functions and Projective Spaces . 53
1.2 Multidimensional Residue Theory . . . . . . . . . . . . 55
1.2.1 Motivations . . . . . . . . . . . . . . . . . . . . 55
1.2.2 Fibre Bundles and Induced Homomorphisms on
(Co)homology . . . . . . . . . . . . . . . . . . . 59
1.2.3 The Bundle Structure on a Tubular Neighbour-
hood of a Manifold . 65
1.2.4 Definition of Residue 67
1.2.5 Leray's Residues 72
v
vi CONTENTS
1.3 Function Spaces . . . . . . . . . . . . . . 74
1.3.1 Motivations and Basic Definitions 75
1.3.2 Properties of the Function Spaces 77
1. 4 Parameter-Dependent Integrals 84
1.4.1 General Remarks ........ . 84
1.4.2 Examples . . . . . . . . . . . . . 85
1.4.3 Thorn's Triviality Theorem (Isotopy Lemma) . 91
1.4.4 Ramified Homology Classes . . . . . . . . . . 95
1.4.5 Differentiation Formula for Parameter-Dependent
Integrals . . . . . . . . . . . . . . . . . . . 101
1.5 Homogeneous Functions and Projective Spaces . . 104
1.5.1 General Remarks . . . . . . . . . . . . . . 104
1.5.2 Projective Spaces . . . . . . . . . . . . . . 105
1.5.3 Holomorphic Forms on Projective Spaces. . 107
1.6 Integration on Fibre Bundles . . . . . . . . . . . . 113
1.6.1 Motivations . . . . . . . . . . . . . . . . . . 113
1.6.2 Relative Spectral Sequence on Homology . . 124
1.6.3 Construction of the Boundary Homomorphism .. 127
1. 6 .4 Le ray ' s H omomorp h.1 sms m. EP2 ' q . . • • • . . • • . 128
1.6.5 Relative Spectral Sequence on Cohomology and
Duality . . . . . . . . . . . . . . 130
1.6.6 Definition of Repeated Integrals . 132
2 Symplectic and Contact Structures 137
2.1 Main Definitions ...................... 137
2.1.1 Preliminary Remarks ................ 137
2.1.2 Homogeneous Symplectic Structure and Contact
Structure . . . . . . . . . . . . . . . . . . 140
2.2 Symplectization and Contactization . . . . . . . . 143
2.2.1 Contactization of a Symplectic Structure . 143
2.2.2 Symplectization of a Contact Structure . . 145
2.2.3 The Contact Product of Contact Structures . 148
2.3 Symplectic and Contact Transformations . . . . . . . 150
2.3.1 Contact and Symplectic Biholomorphisms . . 150
2.3.2 Hamiltonian Functions. Contact and Hamilto-
nian Fields . . . . . . . . . . . . . . . . . 152
2.3.3 Hamiltonian and Contact Distributions ...... 153
CONTENTS vii
2.3.4 Computational Formulas. Examples . . 156
2.4 Lagrangian and Legendre Manifolds . . 160
2.4.1 General Remarks . . . . . . . . . . 160
2.4.2 The Basic Definitions . . . . . . . . 161
2.4.3 Nonsingular Legendre Manifolds . . 163
2.4.4 Representation of Arbitrary Legendre Manifolds . 166
2.4.5 Classification Lemma ................ 171
2.5 Geometric Representation of Contact Transformations . 174
2.5.1 Nonsingular Transformations . . . . . . . . . . 174
2.5.2 Representation of Arbitrary Transformations . . 179
2.5.3 The Legendre Transform of an Analytfc Set . . 181
2.5.4 The Composition of Contact Transformations . 190
3 Integral Transformations of Ramified Analytic Functions195
3.1 Integral Representations Associated with Contact Trans
formations . . . . . . . . . . . . . . . . . 195
3.1.1 Main Definitions ....... . . 195
3.1.2 Special Integral Representation . 203
3.1.3 General Integral Representation . 208
3.2 Integral Transformations of Homogeneous Functions . . 212
3.2.1 Definition of the F-Transformation . . . . . . . 212
3.2.2 Action of F-Transformation in Function Spaces.
The Invertibility Theorem . 218
3.2.3 Commutation Formulas . . . . . 223
3.3 The R-Transformation . . . . . . . . . . 229
3.3.1 Definition and Basic Properties . 229
3.3.2 Asymptotic Expansion of the R-Transform of An
alytic Function . . . . . . . . . . . . . . 233
3.4 a1 a ~-Transformation . . . . . . . . . . . . . . . . 239
3.4.1 Definition and Commutation Formulas . 239
3.4.2 Symmetrization of aI 8~-Transformation . 244
4 Laplace-Radon Integral Operators 251
4.1 Canonical Representations of Ramified Functions . 251
4.1.1 General Remarks . 251
4.1.2 Local Theory . 256
4.1.3 Globalization . . . 263
viii CONTENTS
4.2 Laplace-Radon Integral Operators . . . . . . . . . . . . 268
4.2.1 Motivations . . . . . . . . . . . . . . . . . . . . 268
4.2.2 Local Integral Operators and Function Spaces . 273
4.2.3 Global Theory . . . . . . . . . . 277
4.2.4 Pseudodifferential Operators . . 281
4.3 Composition Theorems . . . . . . . . . 283
4.3.1 General Formulas . . . . . . . . 283
4.3.2 Case of Pseudodifferential Operators . 286
5 Cauchy Problem in Spaces of Ramified Functions 289
5.1 Equations with Constant Coefficients . . . . . . . 289
5.1.1 Statement of the Problem . . . . . . . . . . 289
5.1.2 Construction of the Resolving Operator . . . 291
5.1.3 Description of Singularities of the Solution . 295
5.1.4 Examples of Computation of the Singularities . 296
5.1.5 The Inhomogeneous Cauchy Problem. Reduction
to an Ordinary Differential Equation . . . . 300
5.1.6 The Solution of the Reduced Equation . . . 303
5.1.7 Verification of the Consistency Conditions . 309
5.1.8 Propagation of Singularities of Solutions . . 310
5.2 Equations with Variable Coefficients . . . . . . . . 318
5.2.1 The Cauchy Problem with Singular Initial Data . 318
5.2.2 Elementary Solutions . . . . . . . . . . . . 325
5.2.3 Existence of Local Elementary Solution . . 328
5.3 The Characteristic Cauchy Problem . . . . . . . . 334
5.3.1 Statement of the Problem . . . . . . . . . 334
5.3.2 Asymptotic Expansions of the Solution in the
Vicinity of a Noncharacteristic Point ....... 337
5.3.3 The Construction of a Locally Trivial Stratified
Bundle ........................ 341
5.3.4 The Regularizer in the Vicinity of a Characteris-
tic Point . . . . . . . . . . . . . . . . . . . . . . . 354
5.3.5 Estimates for the Leading Term of Asymptotic
Expansion . . . . . . 359
5.3.6 Patching Solutions . . . . . . . . . 368
5.4 Existence Theorems . . . . . . . . . . . . . 368
5.4.1 Equations in a Bounded Domain . 368
CONTENTS ix
5.4.2 A Model Example . . . . . . . . . . . . . . . . . . 378
5.4.3 Equations with Variable Coefficients ....... 381
5.5 The Characteristic Cauchy Problem and Leray's Uni-
formization . . . . . . . . . . . . . 392
5.5.1 Statement of the Problem . . . . . . 392
5.5.2 Uniformization . . . . . . . . . . . . 393
5.5.3 Reduction of the Cauchy Problem . . 397
5.5.4 Investigation of Singularities and Asymptotic Ex-
pansion . . . . . . . . . . . . . . . . . . . . . . . 399
5.5.5 Leray's Uniformization and the 8/8~-Formalism . 401
6 Continuation of Solutions to Elliptic Equations 405
6.1 Statement of the Problem and Review of Some Classical
Results . . . . . . . . . . . . . . . . 405
6.1.1 Motivations ..................... 405
6.1.2 Statement of the Problem ............. 415
6.1.3 The Continuation Problem and Equations in Com-
plex Domain ..................... 418
6.1.4 The Reflection Formula and Continuation of So-
lutions to the Dirichlet Problem . 421
6.2 The Balayage Problem . . . . . . . . . 429
6.2.1 Statement of the Problem . . . . 429
6.2.2 Reduction to Cauchy Problem . . 430
6.2.3 Solution of the Complex Cauchy Problem . 431
6.2.4 Solution of the Balayage Problem . 433
6.2.5 A Counterexample . . . . . . . . . . . . . . 435
6.2.6 A Special Case . . . . . . . . . . . . . . . . 437
6.2. 7 Arbitrary Equations with Constant Coefficients . 441
6.2.8 Examples . . . . . . . . . . . . . . . . . . . . . . 442
6.3 Reflection Formulas and Continuation of Solutions to
Boundary Value Problems . . . . . 453
6.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . 453
6.3.2 Statement of the Problem . . . . . . . . . . . . . 456
6.3.3 Reduction of the Basic Problem to the Problem
with a Predetermined Locus of Singularities of
the Solution . . . . . . . . . . . . . . . 457
6.3.4 The Reflected Fundamental Solution . . . . . . . 462
x CONTENTS
6.3.5 Properties of the Reflected Fundamental Solution 465
6.3.6 Reflection Formula and Remarks . . 467
6,3. 7 Inhomogeneous Reflection Formula . . . . . . . . 4 70
Bibliography 4 75
Index 501