Hyperbolic Manifolds and Holomorphic Mappings: An IntroductionThe 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. |
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 |
| 145 | |
| 147 | |
Other editions - View all
Common terms and phrases
analytic subset assume automorphism Bergman kernel Bergman metric biholomorphic Carathéodory distance Chapter Chern Chern class closed complex compact complex manifold compact Kaehler manifold compact subset complete hyperbolic complete with respect complex manifold complex submanifold complex subspace converges coordinate system Corollary curvature is bounded defined deg f denote distance-decreasing domain in Cn ds² dz dz dz¹ Example finite follows Gaussian curvature group of holomorphic Hence Hermitian manifold holomorphic functions holomorphic mapping holomorphic sectional curvature holomorphic transformations homogeneous hyperbolic complex space hyperbolic manifold inequality integer k-dimensional Kaehler manifold Kobayashi Let f line bundle mapping f Math metric ds minimal model morphic negative constant neighborhood obtain open unit disk polydisk positive number Proof Let Proposition prove punctured disk relatively minimal Ricci tensor Schwarz lemma sectional curvature Siegel domain Theorem 3.1 trivial volume element см