Acknowledgments
Preface to the Second Edition
Preface to the First Edition
0 Principal Ideas of Classical Function Theory
1.A Glimpse of Complex Analysis
2.The Maximum Principle, the Schwarz Lemma, and Applications
3.Normal Families and the Riemann Mapping Theorem
4.Isolated Singularities and the Theorems of Picard
1 Basic Notions of Differential Geometry
0.Introductory Remarks
1.Riemannian Metrics and the Concept of Length
2.Calculus in the Complex Domain
3.Isometrics
4.The Poincare Metric
5.The Schwarz Lemma
6.A Detour into Non-Euclidean Geometry
2 Curvature and Applications
1.Curvature and the Schwarz Lemma Revisited
2.Liouville's Theorem and Other Applications
3.Normal Families and the Spherical Metric
4.A Generalization of Montel's Theorem and the Great Picard Theorem
3 Some New Invariant Metrics
0.Introductory Remarks
1.The Caratheodory Metric
2.The Kobayashi Metric
3.Completeness of the Caratheodory and Kobayashi Metrics
4.An Application of Completeness: Automorphisms
5.Hyperbolicity and Curvature
4 Introduction to the Bergman Theory
0.Introductory Remarks
1.Bergman Basics
2.Invariance Properties of the Bergman Kernel
3.Calculation of the Bergman Kernel
4.About the Bergman Metric
5.More on the Bergman Metric
6.Application to Conformal Mapping
5 A Glimpse of Several Complex Variables
0.Functions of Several Complex Variables
1.Basic Concepts
2.The Automorphism Groups of the Ball and Bidisc.
3.Invariant Metrics and the Inequivalence of the Ball and the Bidisc
Contents
Appendix
1.Introduction
2.Expressing Curvature Intrinsically
3.Curvature Calculations on Planar Domains
Symbols
References
Index
