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Product Code:  GSM/152 
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Hardcover ISBN:  9781470410476 
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Hardcover ISBN:  9781470410476 
Product Code:  GSM/152 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470416874 
Product Code:  GSM/152.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470410476 
eBook ISBN:  9781470416874 
Product Code:  GSM/152.B 
List Price:  $184.00 $141.50 
MAA Member Price:  $165.60 $127.35 
AMS Member Price:  $147.20 $113.20 

Book DetailsGraduate Studies in MathematicsVolume: 152; 2014; 192 ppMSC: Primary 53; 14
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 higherdimensional 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 KählerEinstein metrics, the Bergman kernel expansion due to Tian, Donaldson's lower bound for the Calabi energy, and ArezzoPacard's existence theorem for constant scalar curvature Kähler metrics on blowups.
ReadershipGraduate students and research mathematicians interested in geometric analysis and Kähler geometry.

Table of Contents

Chapters

Chapter 1. Kähler geometry

Chapter 2. Analytic preliminaries

Chapter 3. KählerEinstein metrics

Chapter 4. Extremal metrics

Chapter 5. Moment maps and geometric invariant theory

Chapter 6. Kstability

Chapter 7. The Bergman kernel

Chapter 8. CscK metrics on blowups


Additional Material

Reviews

This is an important book, in a rapidlydeveloping area, that brings the specialist or graduate student working on Kähler geometry to the frontiers of today research. It is not a selfcontained textbook. The prerequisites in geometric invariant theory, for example, would require some devotion from a potential reader grounded on Riemannian geometry; viceversa, a reader broughtup 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


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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 higherdimensional 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 KählerEinstein metrics, the Bergman kernel expansion due to Tian, Donaldson's lower bound for the Calabi energy, and ArezzoPacard's existence theorem for constant scalar curvature Kähler metrics on blowups.
Graduate students and research mathematicians interested in geometric analysis and Kähler geometry.

Chapters

Chapter 1. Kähler geometry

Chapter 2. Analytic preliminaries

Chapter 3. KählerEinstein metrics

Chapter 4. Extremal metrics

Chapter 5. Moment maps and geometric invariant theory

Chapter 6. Kstability

Chapter 7. The Bergman kernel

Chapter 8. CscK metrics on blowups

This is an important book, in a rapidlydeveloping area, that brings the specialist or graduate student working on Kähler geometry to the frontiers of today research. It is not a selfcontained textbook. The prerequisites in geometric invariant theory, for example, would require some devotion from a potential reader grounded on Riemannian geometry; viceversa, a reader broughtup 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