The first edition of this influential book, published in 1970, opened up a completely new field of invariant metrics and hyperbolic manifolds. The large number of papers on the topics covered by the book written since its appearance led Mathematical Reviews to create two new subsections “invariant metrics and pseudo-distances” and “hyperbolic complex manifolds” within the section “holomorphic mappings”. The invariant distance introduced in the first edition is now called the “Kobayashi distance”, and the hyperbolicity in the sense of this book is called the “Kobayashi hyperbolicity” to distinguish it from other hyperbolicities. This book continues to serve as the best introduction to hyperbolic complex analysis and geometry and is easily accessible to students since very little is assumed. The new edition adds comments on the most recent developments in the field.
Author(s): Shoshichi Kobayashi (小林 昭七)
Edition: 2
Publisher: World Scientific
Year: 2005
Language: English
Commentary: Improvements with respect to A696F71D4935D544374FB2208216DAF2 : added bookmarks
Pages: 148
Preface to the New Edition
Preface
Contents
I. The Schwarz Lemma and Its Generalizations
1 The Schwarz-Pick Lemma
2 A Generalization by Ahlfors
3 The Gaussian Plane Minus Two Points
4 Schottky's Theorem
5 Compact Riemann Surfaces of Genus ≧ 2
6 Holomorphic Mappings from an Annulusinto an Annulus
II. Volume Elements and the Schwarz Lemma
1 Volume Element and Associated Hermitian Form
2 Basic Formula
3 Holomorphic Mappings f: M' → M with Compact M'
4 Holomorphic Mappings f: D → M, Where D is a Homogeneous Bounded Domain
5 Affinely Homogeneous Siegel Domains of Second Kind
6 Symmetric Bounded Domains
III. Distance and the Schwarz Lemma
1 Hermitian Vector Bundles and Curvatures
2 The Case Where the Domain is a Disk
3 The Case Where the Domain is a Polydisk
4 The Case Where D is a Symmetric Bounded Domain
IV. Invariant Distances on Complex Manifolds
1 An Invariant Pseudodistance
2 Carathéodory Distance
3 Completeness with Respect to the Carathéodory Distance
4 Hyperbolic Manifolds
5 On Completeness of an Invariant Distance
V. Holomorphic Mappings into Hyperbolic Manifolds
1 The Little Picard Theorem
2 The Automorphism Group of a Hyperbolic Manifold
3 Holomorphic Mappings into Hyperbolic Manifolds
VI. The Big Picard Theorem and Extension of
Holomorphic Mappings
1 Statement of the Problem
2 The Invariant Distance on the Punctured Disk
3 Mappings from the Punctured Disk into a Hyperbolic Manifold
4 Holomorphic Mappings into Compact Hyperbolic Manifolds
5 Holomorphic Mappings into Complete Hyperbolic Manifolds
6 Holomorphic Mappings into Relatively Compact Hyperbolic Manifolds
VII. Generalization to Complex Spaces
1 Complex Spaces
2 Invariant Distances for Complex Spaces
3 Extension of Mappings into Hyperbolic Spaces
4 Normalization of Hyperbolic Complex Spaces
5 Complex V-Manifolds (Now Called Orbifolds)
6 Invariant Distances on M/Γ
VIII. Hyperbolic Manifolds and Minimal Models
1 Meromorphic Mappings
2 Strong Minimality and Minimal Models
3 Relative Minimality
IX. Miscellany
1 Invariant Measures
2 Intermediate Dimensional-Invariant Measures
3 Unsolved Problems
Postscript
1 d_M and c_M
2 Hyperbolicity with Partial Degeneracy
3 Hyperbolicity Criteria
4 Hyperbolic lmbedding
5 Intrinsic Infinitesimal Pseudometrics
6 Self-Mappings
7 Finiteness Theorems
8 Principle for the Construction of d_M
Bibliography
Summary of Notations
Author Index
Subject Index
