Hyperbolic manifolds and holomorphic mappings : an introduction / Shoshichi Kobayashi.

Format:

• Book

Author/Creator:

• Kobayashi, Shōshichi, 1932-2012.

Language:

English

Subjects (All):

• Complex manifolds.
• Holomorphic mappings.

Physical Description:

xii, 148 pages : illustrations ; 24 cm

Edition:

Second edition.

Place of Publication:

Singapore ; Hackensack, NJ : World Scientific, [2005]

Summary:

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 pseudodistances" 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. Book jacket.

Contents:

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

Notes:

• Originally published: New York : M. Dekker, 1970.
• Includes bibliographical references (pages 135-142) and indexes.

ISBN:

• 9812564969
• 9812565892

OCLC:

64273893

Publisher Number:

9789812564962