An introduction to extremal Kähler metrics
A basic problem in differential geometry is to find canonical metrics on manifolds. The best known example of this is the classical uniformization theorem for Riemann surfaces. Extremal metrics were introduced by Calabi as an attempt at finding a higher-dimensional generalization of this result in the setting of Kahler geometry. This book gives an introduction to the study of extremal Kahler metrics and in particular to the conjectural picture relating the existence of extremal metrics on proje…
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- Full title :An introduction to extremal Kähler metrics / Gábor Székelyhidi.
Summary
A basic problem in differential geometry is to find canonical metrics on manifolds. The best known example of this is the classical uniformization theorem for Riemann surfaces. Extremal metrics were introduced by Calabi as an attempt at finding a higher-dimensional generalization of this result in the setting of Kahler geometry. This book gives an introduction to the study of extremal Kahler metrics and in particular to the conjectural picture relating the existence of extremal metrics on projective manifolds to the stability of the underlying manifold in the sense of algebraic geometry. The book addresses some of the basic ideas on both the analytic and the algebraic sides of this picture. An overview is given of much of the necessary background material such as basic Kahler geometry moment maps and geometric invariant theory. Beyond the basic definitions and properties of extremal metrics several highlights of the theory are discussed at a level accessible to graduate students: Yau's theorem on the existence of Kahler-Einstein metrics the Bergman kernel expansion due to Tian Donaldson's lower bound for the Calabi energy and Arezzo-Pacard's existence theorem for constant scalar curvature Kahler metrics on blow-ups. -- Provided by Publisher.
Read More ↓Edition Statements
- Publication Date:©2014
Publication Statements
- Publisher(s):American mathematical Society
- Place of Publication:Providence, Rhode Island
- Language:eng
- Publication Date:©2014
The Physical Object
- Description:xvi, 192 pages ; : 27 cm.
Edition Identifiers
- ISBN:9781470410476, 1470410478
- oclc:ocn871316164
- LC Classification:QA641
- Dewey Decimal:516.3/73
Table of Contents
Kähler Geometry -- Analytic Preliminaries -- Kähler-Einstein Metrics -- Extremal Metrics -- Moment Maps and Geometric Invariant Theory -- K-stability -- The Bergman Kernel -- CscK Metrics on Blow-Ups.
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