Canonical Metrics in Kahler Geometry

Format: Paperback
Pub. Date: 2000-09-01
Publisher(s): Birkhauser
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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
Curvature of Kahler metrics
Extremal Kahler metrics
The space of Kahler metrics
A brief review of Chern classes
Uniformization of Kahler-Einstein manifolds
Calabi-Futaki invariants
Definition of Calabi-Futaki invariants
Localization formula for Calabi-Futaki invariants
Scalar curvature as a moment map
Kahler-Einstein metrics with non-positive scalar curvature
The Calabi-Yau Theorem
Kahler-Einstein metrics for manifolds with c1 (M) < 0
Kahler-Einstein metrics with positive scalar curvature
A variational approach
Existence of Kahler-Einstein metrics
Applications and generalizations
A manifold without Kahler-Einstein metric
K-energy and metrics of constant scalar curvature
Relation to stability
Bibliography 99(2)
Index 101

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