CR Manifolds and the Tangential Cauchy Riemann Complex provides an elementary introduction to CR manifolds and the tangential Cauchy-Riemann Complex and presents some of the most important recent developments in the field. The first half of the book covers the basic definitions and background material concerning CR manifolds, CR functions, the tangential Cauchy-Riemann Complex and the Levi form. The second half of the book is devoted to two significant areas of current research. The first area is the holomorphic extension of CR functions. Both the analytic disc approach and the Fourier transform approach to this problem are presented. The second area of research is the integral kernal approach to the solvability of the tangential Cauchy-Riemann Complex. CR Manifolds and the Tangential Cauchy Riemann Complex will interest students and researchers in the field of several complex variable and partial differential equations.
Author(s): Albert Boggess
Series: Studies in Advanced Mathematics Series
Edition: 1
Publisher: CRC Press
Year: 1991
Language: English
Pages: 382
Front Cover......Page 1
Title......Page 4
Copyright......Page 5
Contents ......Page 6
Introduction ......Page 12
PART I. PRELIMINARIES ......Page 18
1.1 Functions ......Page 19
1.2 Vectors and vector fields ......Page 24
1.3 Forms ......Page 26
1.4 The exterior derivative ......Page 29
1.5 Contractions ......Page 32
2.1 Manifolds ......Page 34
2.2 Submanifolds ......Page 36
2.3 Vectors on manifolds ......Page 40
2.4 Forms on manifolds ......Page 43
2.5 Integration on manifolds ......Page 46
3.1 Complexification of a real vector space ......Page 56
3.2 Complex structures ......Page 58
3.3 Higher degree complexified forms ......Page 62
4.1 The real Frobenius theorem ......Page 68
4.2 The analytic Frobenius theorem ......Page 73
4.3 Almost complex structures ......Page 75
5.1 The spaces D' and E' ......Page 78
5.2 Operations with distributions ......Page 82
5.3 Whitney's extension theorem ......Page 88
5.4 Fundamental solutions for partial differential equations ......Page 91
6.1 Definitions ......Page 96
6.2 Operations with cunents ......Page 101
PART II: CR MANIFOLDS ......Page 112
7.1 Imbedded CR manifolds ......Page 114
7.2 A normal form for a generic CR submanifold ......Page 120
7.3 Quadric submanifolds ......Page 128
7.4 Abstract CR manifolds ......Page 137
8.1 Extrinsic approach - ......Page 139
8.2 Intrinsic approach to C9M ......Page 147
8.3 The equivalence of the extrinsic and intrinsic tangential Cauchy-Riemann complexes ......Page 151
9.1 CR functions ......Page 157
9.2 CR maps ......Page 166
10.1 Definitions ......Page 173
10.2 The Levi form for an imbedded CR manifold ......Page 176
10.3 The Levi form of a real hypersurface ......Page 180
11.1 The real analytic imbedding theorem ......Page 186
11.2 Nirenberg 's nonimbeddable example ......Page 189
12.1 Bloom?raham normal form ......Page 196
12.2 Rigid and semirigid submanifolds ......Page 200
12.3 More on the Levi form ......Page 202
12.5 Nongeneric and non-CR manifolds ......Page 204
PART III: THE HOLOMORPHIC EXTENSION OF CR FUNCTiONS ......Page 206
13 An Approximation Theorem ......Page 208
14.1 Lewy's CR extension theorem for hypersurfaces ......Page 215
14.2 The CR extension theorem for higher codimension ......Page 217
14.3 Examples ......Page 219
15 The Analytic Disc Technique ......Page 223
15.1 Reduction to analytic discs ......Page 224
15.2 Analytic discs for hypersurfaces ......Page 225
15.3 Analytic discs for quadric submanifolds ......Page 227
15.4 Bishop's equation ......Page 231
15.5 The proof of the analytic disc theorem for the general case ......Page 238
16 The Fourier Transform Technique ......Page 246
16.1 A Fourier inversion formula ......Page 247
16.2 The hypoanalytic wave front set ......Page 254
16.3 The hypoanalytic wave front set and the Levi form ......Page 261
17.1 The Fourier integral approach in the nonrigid case ......Page 268
17.2 The holomorphic extension of CR distributions ......Page 271
17.3 CR extension near points of higher type ......Page 274
17.4 Analytic hypoellipticity ......Page 277
PART IV: SOLVABILITY OF THE TANGENTIAL CAUCHY-RIEMANN COMPLEX ......Page 280
18.1 Definitions ......Page 282
18.2 A homotopy formula ......Page 289
19 Fundamental Solutions for the Exterior Derivative and Cauchy-Riemann Operators ......Page 294
19.1 Fundamental solutions for d on R^n ......Page 295
19.2 Fundamental solutions for \bar\partia on C^n ......Page 298
19.3 Bochner's global CR extension theorem ......Page 308
20.1 A general class of kernels ......Page 311
20.2 A formal identity ......Page 314
20.3 The solution to the cauchy-riemann equations on a convex domain ......Page 316
20.4 Boundary value results for Henkin's kernels ......Page 320
21.1 The first fundamental solution for the tangential cauchy-riemann complex ......Page 329
21.2 A second fundamental solution to the tangential cauchy-riemann complex ......Page 334
22 A Local Solution to the Tangential cauchy-riemann Equations ......Page 344
23.1 ans Lewy's nonsolvability example ......Page 351
23.2 Henkin's criterion for local solvability at the top degree ......Page 354
24.1 More on the Bochner-Martinelli kernel ......Page 359
24.2 Kernels for strictly pseudoconvex boundaries ......Page 362
24.4 Wealdy convex boundaries ......Page 365
24.5 Solvability of the tangential cauchy-riemann complex in other geometries ......Page 366
Bibliography ......Page 371
Notation ......Page 376
Index ......Page 378
Back Cover......Page 382
