Canonical Metrics in Kahler Geometry

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Format: Paperback
Pub. Date: 2000-09-01
Publisher(s): Birkhauser
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Summary

There has been fundamental progress in complex differential geometry in the last two decades. For one, the uniformization theory of canonical Ka'hler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. The aim of this monograph is to give an essentially self-contained introduction to the theory of canonical Ka'hler metrics on complex manifolds. It also presents the reader with some advanced topics in complex differential geometry not easily found elsewhere. The topics include Calabi-Futaki invariants, extremal Ka'hler metrics, the Calabi-Yau theorem on existence of Ka'hler Ricci-flat metrics, and recent progress on Ka'hler-Einstein metrics with positive scalar curvature. Applications of Ka'hler-Einstein metrics to the uniformization theory are also discussed. Readers with a good general knowledge of differential geometry and partial differential equations should be able to grasp and appreciate the materials in this monograph.

Table of Contents

Preface vii
Introduction to Kahler manifolds
Kahler metrics
1(3)
Curvature of Kahler metrics
4(7)
Extremal Kahler metrics
The space of Kahler metrics
11(3)
A brief review of Chern classes
14(4)
Uniformization of Kahler-Einstein manifolds
18(5)
Calabi-Futaki invariants
Definition of Calabi-Futaki invariants
23(6)
Localization formula for Calabi-Futaki invariants
29(6)
Scalar curvature as a moment map
35(64)
Kahler-Einstein metrics with non-positive scalar curvature
The Calabi-Yau Theorem
43(12)
Kahler-Einstein metrics for manifolds with c1 (M) < 0
55(3)
Kahler-Einstein metrics with positive scalar curvature
A variational approach
58(4)
Existence of Kahler-Einstein metrics
62(23)
Examples
85(4)
Applications and generalizations
A manifold without Kahler-Einstein metric
89(4)
K-energy and metrics of constant scalar curvature
93(3)
Relation to stability
96(3)
Bibliography 99(2)
Index 101

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