Differential Geometry and Analysis on CR Manifolds (eBook)

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