Invariant Distances and Metrics in Complex Analysis
| Date: 27 April 2011, 08:26
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Contents
Preface
I Hyperbolic geometry of the unit disc 1
Exercises 14
II The Carathr odory pseudodistance and the Carathdodory-Reiffen pseudometric 15
2.1 Definitions. General Schwarz-Pick Lemma 16
2.2 Balanced domains 18
2.3 Caratheodory hyperbolicity 27
2.4 The Caratheodory topology 29
2.5 Properties of c('f and y. Length of curve. Inner Caratheodory pseudodistance 33
2.6 Two applications 48
2.7 A class of n-circled domains 53
Notes 65
Exercises 66
III The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric 71
3.1 The Lempert function and the Kobayashi pseudodistance 71
3.2 Tautness 77
3.3 General properties of k 82
3.4 An extension theorem 87
3.5 The Kobayashi-Royden pseudometric 90
3.6 The Kobayashi-Buseman pseudometric 99
3.7 Product-formula 106
Notes 108
Exercises 109
IV Contractible systems 111
4.1 Abstract point of view 111
4.2 Extremal problems for plurisubharmonic functions 115
4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics. C' -pseudodistances 139
4.4 Example - elementary n-circled domains 149
Notes 152
Exercises 153
V Contractible functions and metrics for the annulus 154
Notes 165
Exercises 166
VI The Bergman metric 169
6.1 The Bergman kernel 169
6.2 The Bergman pseudometric 185
6.3 Comparison and localization 190
6.4 The Skwarczynski pseudometric 195
Notes 198
Exercises 200
VII Hyperbolicity and completeness 202
7.1 Global hyperbolicity 202
7.2 Local hyperbolicity 207
7.3 Completeness - general discussion 213
7.4 Carathdodory completeness 216
7.5 Kobayashi completeness 223
7.6 Bergman completeness 230
Notes 234
Exercises 235
VIII Complex geodesics. Lempert's theorem 237
8.1 Complex geodesics 237
8.2 Lempert's theorem 243
8.3 Uniqueness of complex geodesics 255
8.4 Geodesics in convex complex ellipsoids 264
8.5 Biholomorphisms of complex ellipsoids 278
8.6 Schwarz Lemma - the case of equality 281
8.7 Criteria for biholomorphicity 285
Notes 288
Exercises 290
IX Product-property 296
Exercises 309
X Comparison on strongly pseudoconvex domains 310
10.1 Strongly pseudoconvex domains 311
10.2 The boundary behavior of the Carathdodory and the Kobayashi distances 316
10.3 Localization 326
10.4 Boundary behavior of the Caratheodory-Reiffen and the Kobayashi-Royden metrics 331
10.5 A comparison of distances 342
10.6 Characterization of the unit ball by its automorphism group 344
Notes 352
Exercises 353
Miscellanea 355
A The automorphism group of bounded domains 355
B Holomorphic curvature 356
C Complex geodesics 359
D Criteria for biholomorphicity 361
E Boundary behavior of contractible metrics on weakly pseudoconvex domains 363
Appendix 367
HF Holomorphic functions 367
PSH Subharmonic and plurisubharmonic functions 370
PSC Domains of holomorphy and pseudoconvex domains 375
AUT Automorphisms 379
Automorphisms of the unit disc 379
Automorphisms of the unit polydisc 379
Automorphisms of the unit Euclidean ball 380
GR Green function and Dirichlet problem 380
MA Monge-Ampere operator 383
H Hardy spaces 384
References 387
List of symbols 400
Index 405
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