Invariant distances and metrics in complex analysis

Preface......Page 8
Contents......Page 10
I Hyperbolic geometry of the unit disc......Page 14
Exercises......Page 27
II The Carathr odory pseudodistance and the Carathdodory-Reiffen pseudometric......Page 28
2.1 Definitions. General Schwarz-Pick Lemma......Page 29
2.2 Balanced domains......Page 31
2.3 Caratheodory hyperbolicity......Page 40
2.4 The Caratheodory topology......Page 42
2.5 Properties of c(\'f and y. Length of curve. Inner Caratheodory pseudodistance......Page 46
2.6 Two applications......Page 61
2.7 A class of n-circled domains......Page 66
Notes......Page 78
Exercises......Page 79
3.1 The Lempert function and the Kobayashi pseudodistance......Page 84
3.2 Tautness......Page 90
3.3 General properties of k......Page 95
3.4 An extension theorem......Page 100
3.5 The Kobayashi-Royden pseudometric......Page 103
3.6 The Kobayashi-Buseman pseudometric......Page 112
3.7 Product-formula......Page 119
Notes......Page 121
Exercises......Page 122
4.1 Abstract point of view......Page 124
4.2 Extremal problems for plurisubharmonic functions......Page 128
4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics. C\' -pseudodistances......Page 152
4.4 Example - elementary n-circled domains......Page 162
Notes......Page 165
Exercises......Page 166
V Contractible functions and metrics for the annulus......Page 167
Notes......Page 178
Exercises......Page 179
6.1 The Bergman kernel......Page 182
6.2 The Bergman pseudometric......Page 198
6.3 Comparison and localization......Page 203
6.4 The Skwarczynski pseudometric......Page 208
Notes......Page 211
Exercises......Page 213
7.1 Global hyperbolicity......Page 215
7.2 Local hyperbolicity......Page 220
7.3 Completeness - general discussion......Page 226
7.4 Carathdodory completeness......Page 229
7.5 Kobayashi completeness......Page 236
7.6 Bergman completeness......Page 243
Notes......Page 247
Exercises......Page 248
8.1 Complex geodesics......Page 250
8.2 Lempert\'s theorem......Page 256
8.3 Uniqueness of complex geodesics......Page 268
8.4 Geodesics in convex complex ellipsoids......Page 277
8.5 Biholomorphisms of complex ellipsoids......Page 291
8.6 Schwarz Lemma - the case of equality......Page 294
8.7 Criteria for biholomorphicity......Page 298
Notes......Page 301
Exercises......Page 303
IX Product-property......Page 309
Exercises......Page 322
X Comparison on strongly pseudoconvex domains......Page 323
10.1 Strongly pseudoconvex domains......Page 324
10.2 The boundary behavior of the Carathdodory and the Kobayashi distances......Page 329
10.3 Localization......Page 339
10.4 Boundary behavior of the Caratheodory-Reiffen and the Kobayashi-Royden metrics......Page 344
10.5 A comparison of distances......Page 355
10.6 Characterization of the unit ball by its automorphism group......Page 357
Notes......Page 365
Exercises......Page 366
A The automorphism group of bounded domains......Page 368
B Holomorphic curvature......Page 369
C Complex geodesics......Page 372
D Criteria for biholomorphicity......Page 374
E Boundary behavior of contractible metrics on weakly pseudoconvex domains......Page 376
HF Holomorphic functions......Page 380
PSH Subharmonic and plurisubharmonic functions......Page 383
PSC Domains of holomorphy and pseudoconvex domains......Page 388
Automorphisms of the unit polydisc......Page 392
GR Green function and Dirichlet problem......Page 393
MA Monge-Ampere operator......Page 396
H Hardy spaces......Page 397
References......Page 400
List of symbols......Page 413
Index......Page 418