Geometric Analysis of the Bergman Kernel and Metric

Preface......Page 8
Contents......Page 12
1.1 The Bergman Kernel......Page 15
1.1.1 Calculating the Bergman Kernel......Page 28
1.1.2 The Poincaré-Bergman Distance on the Disc......Page 33
1.1.3 Construction of the Bergman Kernel by Way of Differential Equations......Page 34
1.1.4 Construction of the Bergman Kernel by Way of Conformal Invariance......Page 37
1.2 The Szegő and Poisson–Szegő Kernels......Page 39
1.3 Formal Ideas of Aronszajn......Page 49
1.4 A New Bergman Basis......Page 50
1.5 Further Examples......Page 53
1.6 A Real Bergman Space......Page 54
1.7 The Behavior of the Singularity in a General Setting......Page 55
1.8 The Annulus......Page 57
1.9.2 The Case of the Disc......Page 59
1.9.3 The Unit Ball in Cn......Page 63
1.9.4 Strongly Pseudoconvex Domains......Page 66
1.10 Multiply Connected Domains......Page 67
1.11 The Bergman Kernel for a Sobolev Space......Page 68
1.12 Ramadanov\'s Theorem......Page 70
1.13 Coda on the Szegő Kernel......Page 72
1.14 Boundary Localization......Page 73
1.14.2 A Representative Result......Page 74
1.14.4 Domains in Higher-Dimensional Complex Space......Page 76
Exercises......Page 79
2.1 Smoothness to the Boundaryof Biholomorphic Mappings......Page 84
2.2 Boundary Behavior of the Bergman Metric......Page 94
2.3 The Biholomorphic Inequivalence of the Ball and the Polydisc......Page 96
Exercises......Page 97
3.1 Bergman Representative Coordinates......Page 99
3.2.1 Preliminary Remarks......Page 102
3.2.2 Introduction to the Poisson–Bergman Kernel......Page 103
3.2.3 Boundary Behavior......Page 106
3.3 Ideas of Fefferman......Page 110
3.4 Results on the Invariant Laplacian......Page 112
3.5 Dirichlet Problem for Invariant Laplacian......Page 121
Exercises......Page 127
4.2 Zonal Harmonics......Page 129
4.3 The Complex Domain......Page 142
4.4 An Application to the Bergman Projection......Page 153
Exercises......Page 157
5.1 Introductory Remarks......Page 158
5.2 Semicontinuity of Automorphism Groups......Page 162
5.3.1 Finite Type in Dimension Two......Page 167
5.4 The Semicontinuity Theorem......Page 177
5.6 Further Remarks......Page 179
5.7 The Lu Qi-Keng Conjecture......Page 180
5.8 The Lu Qi-Keng Theorem......Page 182
5.9 The Dimension of the Bergman Space......Page 185
5.10.1 Kernel Forms......Page 189
5.10.2 The Invariant Metric......Page 193
5.11 Boundary Behavior of the Bergman Metric......Page 195
Exercises......Page 196
6.1 The Diederich–Fornæss Worm Domain......Page 198
6.2 More on the Worm......Page 203
6.3 Non-Smooth Versions of the Worm Domain......Page 210
6.4 Irregularity of the Bergman Projection......Page 211
6.5 Irregularity of the Bergman Kernel......Page 216
6.6 The Kohn Projection Formula......Page 218
6.7 Boundary Behavior of the Bergman Kernel......Page 219
6.7.1 Hörmander\'s Result on Boundary Behavior......Page 220
6.7.2 The Fefferman\'s Asymptotic Expansion......Page 226
6.8 The Bergman Kernel for a Sobolev Space......Page 232
6.9 Regularity of the Dirichlet Problem......Page 235
6.10.1 Introduction......Page 239
6.11 Proof of Theorem 6.10.1......Page 240
6.12 Application of the Monge–Ampère Equation......Page 244
6.13 An Example of David Barrett......Page 246
6.14 The Bergman Kernel as a Hilbert Integral......Page 256
Exercises......Page 260
7.1 What is the Scaling Method?......Page 262
7.2.1 Nonisotropic Scaling......Page 263
7.2.2 Normal Convergence of Sets......Page 265
7.2.3 Localization......Page 266
7.3.1 The Main Goal......Page 272
7.3.2 The Bergman Metric near Strictly Pseudoconvex Boundary Points......Page 273
Exercises......Page 274
8 Concluding Remarks......Page 283
Table of Notation......Page 284
Bibliography......Page 286
Index......Page 295