Extremal Kaehler Metrics and Kaehler-Ricci Flow

2 March 2008 - 29 March 2008

Aims

This period is in part conceived as a sequel to the programme `Differential Geometry and Topology' that took place at the CRM in 2004. Attention will focus on two fast-developing and inter-related topics from that meeting.

The theory of extremal Kaehler metrics has its origin in the problem of finding the nicest possible metrics on a manifold. Calabi's approach was to consider metrics defined as crtical points for the L2 norm of the scalar curvature in a given Kaehler class. They include constant scalar curvature (CSC) Kaehler metrics and Kaehler-Einstein metrics. Currect work explores ways in which the existence of extremal Kaehler metrics is equivalent to some kind of stability, in the spirit of Geometric Invariant Theory. Coverage will include the study of extremal metrics on toric surfaces using convex functions, and numerical methods.

Ricci flow has proved spectacularly successful in generating metrics with constant curvature in dimensions 3 and above. This theory adapts well to the theory of complex and Kaehler manifolds, and provides a way of establishing the existence of Kaehler-Einstein metrics according to properties of the first Chern class. Currect work investigates the convergence of Kaehler-Ricci and related flows under various stability hypotheses. It can be used to decide whether a particular Kaehler manifold is Stein, and provide approximations using Bergman metrics. Of particular interest are Kaehler-Ricci solitons, that can arise as limiting solutions and generalize Einstein metrics.