Canonical Metrics in Kähler Geometry / Edition 1

by Gang Tian, M. Akveld

Overview

There has been fundamental progress in complex differential geometry in the last two decades. For one, The uniformization theory of canonical Kähler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. This monograph gives an introduction to the theory of canonical Kähler metrics on complex manifolds. It also presents some advanced topics not easily found elsewhere.

Product Details

ISBN-13:	9783764361945
Publisher:	Birkhäuser Basel
Publication date:	10/02/2000
Series:	Lectures in Mathematics. ETH Zürich
Edition description:	2000
Pages:	101
Product dimensions:	7.01(w) x 10.00(h) x 0.28(d)

Table of Contents

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.

Editorial Reviews

"This little monograph offers an essentially self-contained introduction to the theory of canonical Kähler metrics on complex manifolds. …The author presents some advanced topics which are hard [to] find elsewhere… This graduate course on Kähler-Einstein metrics can be recommended to all those interested in recent developments within complex differential geometry."

—Publicationes Mathematicae

"This monograph includes an essentially self-contained introduction to the theory of canonical Kähler metrics on complex manifolds."

—Zentralblatt Math

From the Publisher