Invariant Distances and Metrics in Complex Analysis

✍ Scribed by Marek Jarnicki; Peter Pflug

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 420
• Series: De Gruyter Expositions in Mathematics; 9
• Category: Library

✦ Table of Contents

Preface
I Hyperbolic geometry of the unit disc
Exercises
II The Carathéodory pseudodistance and the Carathéodory-Reiffen pseudometric
2.1 Definitions. General Schwarz-Pick Lemma
2.2 Balanced domains
2.3 Carathéodory hyperbolicity
2.4 The Carathéodory topology
2.5 Properties of c(*)and γ. Length of curve. Inner Carathéodory pseudodistance
2.6 Two applications
2.7 A class of n-circled domains
Notes
Exercises
III The Kobayashi pseudodistance and the Kobayashi-Royden pseudometric
3.1 The Lempert function and the Kobayashi pseudodistance
3.2 Tautness
3.3 General properties of k
3.4 An extension theorem
3.5 The Kobayashi-Royden pseudometric
3.6 The Kobayashi-Buseman pseudometric
3.7 Product-formula
Notes
Exercises
IV Contractible systems
4.1 Abstract point of view
4.2 Extremal problems for plurisubharmonic functions
4.3 Inner pseudodistances. Integrated forms. Derivatives. Buseman pseudometrics. C1-pseudodistances
4.4 Example – elementary n-circled domains
Notes
Exercises
V Contractible functions and metrics for the annulus
Notes
Exercises
VI The Bergman metric
6.1 The Bergman kernel
6.2 The Bergman pseudometric
6.3 Comparison and localization
6.4 The Skwarczyński pseudometric
Notes
Exercises
VII Hyperbolicity and completeness
7.1 Global hyperbolicity
7.2 Local hyperbolicity
7.3 Completeness – general discussion
7.4 Carathéodory completeness
7.5 Kobayashi completeness
7.6 Bergman completeness
Notes
Exercises
VIII Complex geodesics. Lempert’s theorem
8.1 Complex geodesics
8.2 Lempert’s theorem
8.3 Uniqueness of complex geodesies
8.4 Geodesics in convex complex ellipsoids
8.5 Biholomorphisms of complex ellipsoids
8.6 Schwarz Lemma – the case of equality
8.7 Criteria for biholomorphicity
Notes
Exercises
IX Product-property
Exercises
X Comparison on strongly pseudoconvex domains
10.1 Strongly pseudoconvex domains
10.2 The boundary behavior of the Carathéodory and the Kobayashi distances
10.3 Localization
10.4 Boundary behavior of the Carathéodory-Reiffen and the Kobayashi-Royden metrics
10.5 A comparison of distances
10.6 Characterization of the unit ball by its automorphism group
Notes
Exercises
Miscellanea
A The automorphism group of bounded domains
B Holomorphic curvature
C Complex geodesics
D Criteria for biholomorphicity
E Boundary behavior of contractible metrics on weakly pseudoconvex domains
Appendix
HF Holomorphic functions
PSH Subharmonic and plurisubharmonic functions
PSC Domains of holomorphy and pseudoconvex domains
AUT Automorphisms
Automorphisms of the unit disc
Automorphisms of the unit polydisc
Automorphisms of the unit Euclidean ball
GR Green function and Dirichlet problem
MA Monge-Ampère operator
H Hardy spaces
References
List of symbols
Index