An Introduction To Extremal Kahler Metrics (graduate Studies In Mathematics)
by Gabor Szekelyhidi /
2014 / English / PDF
14.7 MB Download
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.
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.
