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Canonical Metrics in Kähler Geometry

Canonical Metrics in Kähler Geometry
Catalogue Information
Field name Details
Dewey Class 516.36
Title Canonical Metrics in Kähler Geometry ([EBook]) / by Gang Tian.
Author Tian, Gang
Other name(s) SpringerLink (Online service)
Publication Basel : Birkhäuser , 2000.
Physical Details VII, 101 pages : online resource.
Series Lectures in Mathematics ETH Zürich, Department of Mathematics Research Institute of Mathematics
ISBN 9783034883894
Summary Note There has been fundamental progress in complex differential geometry in the last two decades. For one, the uniformization theory of canonical Khler 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 Khler 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 Khler metrics, the Calabi-Yau theorem on existence of Khler Ricci-flat metrics, and recent progress on Khler-Einstein metrics with positive scalar curvature. Applications of Khler-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.:
Contents note 1 Introduction to Kähler manifolds -- 1.1 Kähler metrics -- 1.2 Curvature of Kähler metrics -- 2 Extremal Kähler metrics -- 2.1 The space of Kähler metrics -- 2.2 A brief review of Chern classes -- 2.3 Uniformization of Kähler-Einstein manifolds -- 3 Calabi-Futaki invariants -- 3.1 Definition of Calabi-Futaki invariants -- 3.2 Localization formula for Calabi-Futaki invariants -- 4 Scalar curvature as a moment map -- 5 Kähler-Einstein metrics with non-positive scalar curvature -- 5.1 The Calabi-Yau Theorem -- 5.2 Kähler-Einstein metrics for manifolds with c1(M) < 0 -- 6 Kähler-Einstein metrics with positive scalar curvature -- 6.1 A variational approach -- 6.2 Existence of Kähler-Einstein metrics -- 6.3 Examples -- 7 Applications and generalizations -- 7.1 A manifold without Kähler-Einstein metric -- 7.2 K-energy and metrics of constant scalar curvature -- 7.3 Relation to stability.
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