**Graduate Studies in Mathematics**

Volume: 152;
2014;
192 pp;
Hardcover

MSC: Primary 53; 14;

**Print ISBN: 978-1-4704-1047-6
Product Code: GSM/152**

List Price: $61.00

AMS Member Price: $48.80

MAA Member Price: $54.90

**Electronic ISBN: 978-1-4704-1687-4
Product Code: GSM/152.E**

List Price: $57.00

AMS Member Price: $45.60

MAA Member Price: $51.30

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#### Supplemental Materials

# An Introduction to Extremal Kähler Metrics

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*Gábor Székelyhidi*

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
Kähler geometry.

This book gives an introduction to the study of extremal Kähler
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 Kähler 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 Kähler-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 Kähler
metrics on blow-ups.

#### Readership

Graduate students and research mathematicians interested in geometric analysis and Kähler geometry.

#### Reviews & Endorsements

This is an important book, in a rapidly-developing area, that brings the specialist or graduate student working on Kähler geometry to the frontiers of today research. It is not a self-contained textbook. The pre-requisites in geometric invariant theory, for example, would require some devotion from a potential reader grounded on Riemannian geometry; vice-versa, a reader brought-up in algebraic geometry would have to make an effort to follow the part on analysis or differential geometry. The rewards for these efforts justify everything: the book is well organized, and when it sketches an argument, there are precise pointers to the literature for full details.

-- MAA Reviews

Very well written, the book provides a survey of extremal ahler metrics and it promotes to tackle other topics.

-- Zentralblatt Math

#### Table of Contents

# Table of Contents

## An Introduction to Extremal Kahler Metrics

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Introduction xiii14 free
- Kähler geometry 118 free
- Analytic preliminaries 2340
- Kähler-Einstein metrics 3552
- Extremal metrics 5774
- Moment maps and geometric invariant theory 85102
- K-stability 105122
- The Bergman kernel 129146
- CscK metrics on blow-ups 153170
- Bibliography 185202
- Index 191208 free
- Back Cover Back Cover1210