An Introduction to Extremal Kahler Metrics

Gabor Szekelyhidi (Autor)

Buch | Hardcover

192 Seiten

2014
American Mathematical Society (Verlag)
978-1-4704-1047-6 (ISBN)

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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 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.

Gabor Szekelyhidi, University of Notre Dame, IN.

Kahler geometry
Analytic preliminaries
Kahler-Einstein metrics
Extremal metrics
Moment maps and geometric invariant theory K-stability
The Bergman kernel CscK metrics on blow-ups
Bibliography
Index

Reihe/Serie	Graduate Studies in Mathematics
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	520 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-1047-8 / 1470410478
ISBN-13	978-1-4704-1047-6 / 9781470410476
Zustand	Neuware