导语

  

内容提要

  

    本书获得1994年美国数学协会Beckenbach图书奖！
    在这本经典著作的第二版中，Steven Krantz扩充了有关经典非欧几何的内容。他展示了如何从复圆盘的不变几何中，以一种自然的方式发展非欧几何。他还介绍了Bergman核和度量，给出深刻的应用，其中一些从未出版过。
    总的来说，在第一版成功的基础上，新版做了大量的修改润色和重新思考。这是第一本也是唯一一本描述Ahlfors著名思想的内容、背景、细节和应用的书，其中包括曲率、Schwarz引理及其在复分析中的应用。
    本书从零开始，只需要很少的复变理论背景知识，就将读者带入当前活跃的研究领域。这些领域包括a) Carathéodory和Kobayashi度量，b) Bergman核和度量，c) 共形映射的边界延拓。此外，本书还介绍了多复变理论，讨论并证明了Poincaré关于球与多圆盘的不双全纯等价的著名定理。

Acknowledgments
Preface to the Second Edition
Preface to the First Edition
0  Principal Ideas of Classical Function Theory
  1.A Glimpse of Complex Analysis
  2.The Maximum Principle, the Schwarz Lemma, and Applications
  3.Normal Families and the Riemann Mapping Theorem
  4.Isolated Singularities and the Theorems of Picard
1  Basic Notions of Differential Geometry
  0.Introductory Remarks
  1.Riemannian Metrics and the Concept of Length
  2.Calculus in the Complex Domain
  3.Isometrics
  4.The Poincare Metric
  5.The Schwarz Lemma
  6.A Detour into Non-Euclidean Geometry
2  Curvature and Applications
  1.Curvature and the Schwarz Lemma Revisited
  2.Liouville's Theorem and Other Applications
  3.Normal Families and the Spherical Metric
  4.A Generalization of Montel's Theorem and the Great Picard Theorem
3  Some New Invariant Metrics
  0.Introductory Remarks
  1.The Caratheodory Metric
  2.The Kobayashi Metric
  3.Completeness of the Caratheodory and Kobayashi Metrics
  4.An Application of Completeness: Automorphisms
  5.Hyperbolicity and Curvature
4  Introduction to the Bergman Theory
  0.Introductory Remarks
  1.Bergman Basics
  2.Invariance Properties of the Bergman Kernel
  3.Calculation of the Bergman Kernel
  4.About the Bergman Metric
  5.More on the Bergman Metric
  6.Application to Conformal Mapping
5  A Glimpse of Several Complex Variables
  0.Functions of Several Complex Variables
  1.Basic Concepts
  2.The Automorphism Groups of the Ball and Bidisc.
  3.Invariant Metrics and the Inequivalence of the Ball and the Bidisc
  Contents
Appendix
  1.Introduction
  2.Expressing Curvature Intrinsically
  3.Curvature Calculations on Planar Domains
Symbols
References
Index