Complex Monge Amp Re Equations And Geodesics In The Space Of K Hler Metrics

Emerging Trends in Visual Computing

This book is an outcome of the LIX Fall Colloquium on the Emerging Trends in Visual Computing, ETVC 2008, which was held in Palaiseau, France, November 18-20, 2008. During the event, 25 renowned invited speakers gave lectures on their areas of expertise within the field of visual computing. From these talks, a total of 15 state-of-the-art articles have been assembled in this volume. All articles were thoroughly reviewed and improved, according to the suggestions of the referees. The 15 contributions presented in this state-of-the-art survey are organized in topical sections on: geometric computing, information geometry and applications, computer graphics and vision, information retrieval, and medical imaging and computational anatomy. They are preceded by the abstracts of the talks given at ETVC 2008.

An Introduction to Extremal Kahler 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 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.

Spectral Geometry