Hyperbolic Manifolds And Holomorphic Mappings : An Introduction - Shoshichi Kobayashi

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Hyperbolic Manifolds And Holomorphic Mappings

An Introduction

By: Shoshichi Kobayashi

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Paperback | 7 November 2005 | Edition Number 2

At a Glance

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160 Pages

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Revised

Dimensions(cm)
22.8 x 16.2 x 1.3

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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.

Industry Reviews

"This book continues to serve as a fine introduction to hyperbolic complex analysis at a very elementary level." Zentralblatt MATH "A student with some background in complex differential geometry will find this to be an accessible, yet comprehensive, introduction to the subject." Mathematical Reviews

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Preface to the New Edition	p. vii
Preface	p. ix
The Schwarz Lemma and Its Generalizations	p. 1
The Schwarz-Pick Lemma	p. 1
A Generalization by Ahlfors	p. 3
The Gaussian Plane Minus Two Points	p. 5
Schottky's Theorem	p. 10
Compact Riemann Surfaces of Genus [greater than or equal] 2	p. 11
Holomorphic Mappings from an Annulus into an Annulus	p. 12
Volume Elements and the Schwarz Lemma	p. 16
Volume Element and Associated Hermitian Form	p. 17
Basic Formula	p. 19
Holomorphic Mappings f : M' to M with Compact M'	p. 20
Holomorphic Mappings f : D to M, Where D is a Homogeneous Bounded Domain	p. 25
Affinely Homogeneous Siegel Domains of Second Kind	p. 28
Symmetric Bounded Domains	p. 33
Distance and the Schwarz Lemma	p. 36
Hermitian Vector Bundles and Curvatures	p. 37
The Case Where the Domain is a Disk	p. 40
The Case Where the Domain is a Polydisk	p. 40
The Case Where D is a Symmetric Bounded Domain	p. 41
Invariant Distances on Complex Manifolds	p. 44
An Invariant Pseudodistance	p. 45
Caratheodory Distance	p. 49
Completeness with Respect to the Caratheodory Distance	p. 52
Hyperbolic Manifolds	p. 56
On Completeness of an Invariant Distance	p. 63
Holomorphic Mappings into Hyperbolic Manifolds	p. 67
The Little Picard Theorem	p. 67
The Automorphism Group of a Hyperbolic Manifold	p. 67
Holomorphic Mappings into Hyperbolic manifolds	p. 70
The Big Picard Theorem and Extension of Holomorphic Mappings	p. 77
Statement of the Problem	p. 77
The Invariant Distance on the Punctured Disk	p. 78
Mappings from the Punctured Disk into a Hyperbolic Manifold	p. 81
Holomorphic Mappings into Compact Hyperbolic Manifolds	p. 84
Holomorphic Mappings into Complete Hyperbolic Manifolds	p. 85
Holomorphic Mappings into Relatively Compact Hyperbolic Manifolds	p. 88
Generalization to Complex Spaces	p. 93
Complex Spaces	p. 93
Invariant Distances for Complex Spaces	p. 95
Extension of Mappings into Hyperbolic Spaces	p. 96
Normalization of Hyperbolic Complex Spaces	p. 98
Complex V-Manifolds (Now Called Orbitfolds)	p. 100
Invariant Distances on M/[Gamma]	p. 100
Hyperbolic Manifolds and Minimal Models	p. 103
Meromorphic Mappings	p. 103
Strong Minimality and Minimal Models	p. 104
Relative Minimality	p. 108
Miscellany	p. 115
Invariant Measures	p. 115
Intermediate Dimensional-Invariant Measures	p. 118
Unsolved Problems	p. 125
Postscript	p. 129
Bibliography	p. 135
Summary of Notations	p. 143
Author Index	p. 145
Subject Index	p. 147
Table of Contents provided by Ingram. All Rights Reserved.

Hyperbolic Manifolds And Holomorphic Mappings

An Introduction

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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