Dragomir / Tomassini Differential Geometry and Analysis on CR Manifolds

1. Auflage 2007
ISBN: 978-0-8176-4483-3
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 246, 488 Seiten

Reihe: Progress in Mathematics

Presents many major differential geometric acheivements in the theory of CR manifolds for the first time in book form Explains how certain results from analysis are employed in CR geometry Many examples and explicitly worked-out proofs of main geometric results in the first section of the book making it suitable as a graduate main course or seminar textbook Provides unproved statements and comments inspiring further study

Autoren/Hrsg.

Dragomir, Sorin

Tomassini, Giuseppe

Inhaltsverzeichnis

1;Contents;5
2;Preface;9
3;1 CR Manifolds;16
3.1;1.1 CR manifolds;18
3.1.1;1.1.1 CR structures;18
3.1.2;1.1.2 The Levi form;20
3.1.3;1.1.3 Characteristic directions on nondegenerate CR manifolds;23
3.1.4;1.1.4 CR geometry and contact Riemannian geometry;25
3.1.5;1.1.5 The Heisenberg group;26
3.1.6;1.1.6 Embeddable CR manifolds;29
3.1.7;1.1.7 CR Lie algebras and CR Lie groups;33
3.1.8;1.1.8 Twistor CR manifolds;36
3.2;1.2 The Tanaka–Webster connection;40
3.3;1.3 Local computations;46
3.3.1;1.3.1 Christoffel symbols;47
3.3.2;1.3.2 The pseudo-Hermitian torsion;51
3.3.3;1.3.3 The volume form;58
3.4;1.4 The curvature theory;61
3.4.1;1.4.1 Pseudo-Hermitian Ricci and scalar curvature;65
3.4.2;1.4.2 The curvature forms;66
3.4.3;1.4.3 Pseudo-Hermitian sectional curvature;73
3.5;1.5 The Chern tensor field;75
3.6;1.6 CR structures as G-structures;83
3.6.1;1.6.1 Integrability;85
3.6.2;1.6.2 Nondegeneracy;87
3.7;1.7 The tangential Cauchy–Riemann complex;88
3.7.1;1.7.1 The tangential Cauchy–Riemann complex;88
3.7.2;1.7.3 The Frölicher spectral sequence;99
3.7.3;1.7.4 A long exact sequence;109
3.7.4;1.7.5 Bott obstructions;111
3.7.5;1.7.6 The Kohn–Rossi Laplacian;114
3.8;1.8 The group of CR automorphisms;121
4;2 The Fefferman Metric;124
4.1;2.1 The sub-Laplacian;126
4.2;2.2 The canonical bundle;134
4.3;2.3 The Fefferman metric;137
4.4;2.4 A CR invariant;150
4.5;2.5 The wave operator;155
4.6;2.6 Curvature of Fefferman’s metric;156
4.7;2.7 Pontryagin forms;157
4.8;2.8 The extrinsic approach;162
4.8.1;2.8.1 The Monge–Amp`ere equation;162
4.8.2;2.8.2 The Fefferman metric;165
4.8.3;2.8.3 Obstructions to global embeddability;166
5;3 The CR Yamabe Problem;172
5.1;3.1 The Cayley transform;176
5.2;3.2 Normal coordinates;180
5.3;3.3 A Sobolev-type lemma;191
5.4;3.4 Embedding results;208
5.5;3.5 Regularity results;211
5.6;3.6 Existence of extremals;215
5.7;3.7 Uniqueness and open problems;220
5.8;3.8 The weak maximum principle for b;222
6;4 Pseudoharmonic Maps;226
6.1;4.1 CR and pseudoharmonic maps;227
6.2;4.2 A geometric interpretation;230
6.3;4.3 The variational approach;233
6.4;4.4 Hörmander systems and harmonicity;246
6.4.1;4.4.1 Hörmander systems;248
6.4.2;4.4.2 Subelliptic harmonic morphisms;251
6.4.3;4.4.3 The relationship to hyperbolic PDEs;254
6.4.4;4.4.4 Weak harmonic maps from C(Hn);257
6.5;4.5 Generalizations of pseudoharmonicity;261
6.5.1;4.5.1 The first variation formula;263
6.5.2;4.5.2 Pseudoharmonic morphisms;267
6.5.3;4.5.3 The geometric interpretation of F-pseudoharmonicity;270
6.5.4;4.5.4 Weak subelliptic F-harmonic maps;271
7;5 Pseudo-Einsteinian Manifolds;290
7.1;5.1 The local problem;290
7.2;5.2 The divergence formula;294
7.3;5.3 CR-pluriharmonic functions;295
7.4;5.4 More local theory;309
7.5;5.5 Topological obstructions;311
7.5.1;5.5.1 The first Chern class of T1,0(M);311
7.5.2;5.5.2 The traceless Ricci tensor;314
7.5.3;5.5.3 The Lee class;316
7.6;5.6 The global problem;319
7.7;5.7 The Lee conjecture;327
7.7.1;5.7.1 Quotients of the Heisenberg group by properly discontinuous groups of CR automorphisms;328
7.7.2;5.7.2 Regular strictly pseudoconvex CR manifolds;333
7.7.3;5.7.3 The Bockstein sequence;335
7.7.4;5.7.4 The tangent sphere bundle;336
7.8;5.8 Pseudo-Hermitian holonomy;344
7.9;5.9 Quaternionic Sasakian manifolds;346
7.10;5.10 Homogeneous pseudo-Einsteinian manifolds;356
8;6 Pseudo-Hermitian Immersions;360
8.1;6.1 The theorem of H. Jacobowitz;362
8.2;6.2 The second fundamental form;366
8.3;6.3 CR immersions into Hn+k;371
8.4;6.4 Pseudo-Einsteinian structures;376
8.4.1;6.4.1 CR-pluriharmonic functions and the Lee class;376
8.4.2;6.4.2 Consequences of the embedding equations;379
8.4.3;6.4.3 The first Chern class of the normal bundle;384
9;7 Quasiconformal Mappings;392
9.1;7.1 The complex dilatation;392
9.2;7.2 K-quasiconformal maps;399
9.3;7.3 The tangential Beltrami equations;400
9.3.1;7.3.1 Contact transformations of Hn;403
9.3.2;7.3.2 The tangential Beltrami equation on H1;409
9.4;7.4 Symplectomorphisms;413
9.4.1;7.4.1 Fefferman’s formula and boundary behavior of symplectomorphisms;413
9.4.2;7.4.2 Dilatation of symplectomorphisms and the Beltrami equations;416
9.4.3;7.4.3 Boundary values of solutions to the Beltrami system;418
9.4.4;7.4.4 A theorem of P. Libermann;418
9.4.5;7.4.5 Extensions of contact deformations;420
10;8 Yang–Mills Fields on CR Manifolds;422
10.1;8.1 Canonical S-connections;422
10.2;8.2 Inhomogeneous Yang–Mills equations;425
10.3;8.3 Applications;427
10.3.1;8.3.1 Trivial line bundles;429
10.3.2;8.3.2 Locally trivial line bundles;429
10.3.3;8.3.3 Canonical bundles;431
10.4;8.4 Various differential operators;433
10.5;8.5 Curvature of S-connections;436
11;9 Spectral Geometry;438
11.1;9.1 Commutation formulas;438
11.2;9.2 A lower bound for .1;442
11.2.1;9.2.1 A Bochner-type formula;445
11.2.2;9.2.2 Two integral identities;446
11.2.3;9.2.3 A. Greenleaf’s theorem;449
11.2.4;9.2.4 A lower bound on the .rst eigenvalue of a Folland–Stein operator;454
11.2.5;9.2.5 Z. Jiaqing and Y. Hongcang’s theorem on CR manifolds;456
12;A A Parametrix for;460
13;References;478
14;Index;498