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Differential Geometry and Analysis on CR Manifolds

:	Sorin Dragomir, Giuseppe Tomassini
:	Differential Geometry and Analysis on CR Manifolds
:	Birkhäuser Basel
:	9780817644833
:	1
:	CHF 169.50
:

:	Geometrie
:	English

:	488
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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

	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