Hyperbolic Manifolds and Holomorphic Mappings: An Introduction

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World Scientific, 2005 - Mathematics - 148 pages
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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.
 

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Contents

Chapter I The Schwarz Lemma and Its Generalizations
1
Chapter II Volume Elements and the Schwarz Lemma
17
Chapter IV Invariant Distances on Complex Manifolds
45
Chapter V Holomorphic Mappings into Hyperbolic Manifolds
67
Chapter VI The Big Picard Theorem and Extension of Holomorphic Mappings
77
Chapter VII Generalization to Complex Spaces
93
Chapter VIII Hyperbolic Manifolds and Minimal Models
103
Chapter IX Miscellany
115
Postscript
129
Bibliography
135
Summary of Notations
143
Author Index
145
Subject Index
147
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