Progress in Mathematics

By: Sorin Dragomir, Giuseppe Tomassini

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Hardcover | 17 March 2006

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The study of CR manifolds lie at the intersection of three main mathematical disciplines, partial differential equations, complex analysis in several complex variables, and differential geometry. While the complex analysis and PDEs aspect have been intensly studied in the last fifty years, much effort has been recently made to understand the differential geometric side of the subject.This monograph provides a unified presentation of several differential geometric aspects in the theory of CR manifolds and tangential Cauchy-Riemann equations. It presents topics from the Tanaka-Webster connection, a key contributor to the birth of pseudohermitian geometry, to the major differential geometric acheivements in the theory of CR manifolds, such as Fefferman's metric, pseudo-Einstein structures and the Lee conjecture, CR immersions, subelliptic harmonic maps as a local manifestation of pseudoharmonic maps from a CR manifold, Yang-Mills fields on CR manifolds, to name several. It also aims at explaining how certain results from analysis are employed in CR geometry. Motivated by clear exposition, many examples, explicitly worked out geometric results and stimulating unproved statements and comments referring to the most recent aspects of the theory, this monograph is suitable for researchers and graduate students in differential geometry, complex analysis, and PDEs.

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In fact, it will be invaluable for people working on the differential geometry of CR manifolds. -Thomas Garity, MathSciNet

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Geometry Differential & Riemannian Geometry Complex Analysis

Preface	p. ix
CR Manifolds	p. 1
CR manifolds	p. 3
CR structures	p. 3
The Levi form	p. 5
Characteristic directions on nondegenerate CR manifolds	p. 8
CR geometry and contact Riemannian geometry	p. 10
The Heisenberg group	p. 11
Embeddable CR manifolds	p. 14
CR Lie algebras and CR Lie groups	p. 18
Twistor CR manifolds	p. 21
The Tanaka-Webster connection	p. 25
Local computations	p. 31
Christoffel symbols	p. 32
The pseudo-Hermitian torsion	p. 36
The volume form	p. 43
The curvature theory	p. 46
Pseudo-Hermitian Ricci and scalar curvature	p. 50
The curvature forms [Characters not reproducible]	p. 51
Pseudo-Hermitian sectional curvature	p. 58
The Chern tensor field	p. 60
CR structures as G-structures	p. 68
Integrability	p. 70
Nondegeneracy	p. 72
The tangential Cauchy-Riemann complex	p. 73
The tangential Cauchy-Riemann complex	p. 73
CR-holomorphic bundles	p. 82
The Frolicher spectral sequence	p. 84
A long exact sequence	p. 94
Bott obstructions	p. 96
The Kohn-Rossi Laplacian	p. 99
The group of CR automorphisms	p. 106
The Fefferman Metric	p. 109
The sub-Laplacian	p. 111
The canonical bundle	p. 119
The Fefferman metric	p. 122
A CR invariant	p. 135
The wave operator	p. 140
Curvature of Fefferman's metric	p. 141
Pontryagin forms	p. 142
The extrinsic approach	p. 147
The Monge-Ampere equation	p. 147
The Fefferman metric	p. 150
Obstructions to global embeddability	p. 151
The CR Yamabe Problem	p. 157
The Cayley transform	p. 161
Normal coordinates	p. 165
A Sobolev-type lemma	p. 176
Embedding results	p. 193
Regularity results	p. 196
Existence of extremals	p. 200
Uniqueness and open problems	p. 205
The weak maximum principle for [Delta subscript b]	p. 207
Pseudoharmonic Maps	p. 211
CR and pseudoharmonic maps	p. 212
A geometric interpretation	p. 215
The variational approach	p. 218
Hormander systems and harmonicity	p. 231
Hormander systems	p. 233
Subelliptic harmonic morphisms	p. 236
The relationship to hyperbolic PDEs	p. 239
Weak harmonic maps from C(H[subscript n])	p. 242
Generalizations of pseudoharmonicity	p. 246
The first variation formula	p. 248
Pseudoharmonic morphisms	p. 252
The geometric interpretation of F-pseudoharmonicity	p. 255
Weak subelliptic F-harmonic maps	p. 256
Pseudo-Einsteinian Manifolds	p. 275
The local problem	p. 275
The divergence formula	p. 279
CR-pluriharmonic functions	p. 280
More local theory	p. 294
Topological obstructions	p. 296
The first Chern class of T[subscript 1,0](M)	p. 296
The traceless Ricci tensor	p. 299
The Lee class	p. 301
The global problem	p. 304
The Lee conjecture	p. 312
Quotients of the Heisenberg group by properly discontinuous groups of CR automorphisms	p. 313
Regular strictly pseudoconvex CR manifolds	p. 318
The Bockstein sequence	p. 320
The tangent sphere bundle	p. 321
Pseudo-Hermitian holonomy	p. 329
Quaternionic Sasakian manifolds	p. 331
Homogeneous pseudo-Einsteinian manifolds	p. 341
Pseudo-Hermitian Immersions	p. 345
The theorem of H. Jacobowitz	p. 347
The second fundamental form	p. 351
CR immersions into H[subscript n+k]	p. 356
Pseudo-Einsteinian structures	p. 361
CR-pluriharmonic functions and the Lee class	p. 361
Consequences of the embedding equations	p. 364
The first Chern class of the normal bundle	p. 369
Quasiconformal Mappings	p. 377
The complex dilatation	p. 377
K-quasiconformal maps	p. 384
The tangential Beltrami equations	p. 385
Contact transformations of H[subscript n]	p. 388
The tangential Beltrami equation on H[subscript 1]	p. 394
Symplectomorphisms	p. 398
Fefferman's formula and boundary behavior of symplectomorphisms	p. 398
Dilatation of symplectomorphisms and the Beltrami equations	p. 401
Boundary values of solutions to the Beltrami system	p. 403
A theorem of P. Libermann	p. 403
Extensions of contact deformations	p. 405
Yang-Mills Fields on CR Manifolds	p. 407
Canonical S-connections	p. 407
Inhomogeneous Yang-Mills equations	p. 410
Applications	p. 412
Trivial line bundles	p. 414
Locally trivial line bundles	p. 414
Canonical bundles	p. 416
Various differential operators	p. 418
Curvature of S-connections	p. 421
Spectral Geometry	p. 423
Commutation formulas	p. 423
A lower bound for [lambda subscript 1]	p. 427
A Bochner-type formula	p. 430
Two integral identities	p. 431
A. Greenleaf's theorem	p. 434
A lower bound on the first eigenvalue of a Folland-Stein operator	p. 439
Z. Jiaqing and Y. Hongcang's theorem on CR manifolds	p. 441
A Parametrix for [squ subscript b]	p. 445
References	p. 463
Index	p. 483
Table of Contents provided by Ingram. All Rights Reserved.

Progress in Mathematics

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