Parabolic flows and canonical metrics in Kahler geometry.

Harvard University, Cambridge MA

Investigators

Abstract

The problem of the existence of a constant scalar curvature Kahler metric in a given Kahler class is an important and difficult problem and has provided the motivation for much current research in Kahler geometry. For the special case of Kahler-Einstein metrics on Fano manifolds, existence was conjectured by Yau to be equivalent to the stability of the manifold in the sense of geometric invariant theory. The principal investigator proposes to study three parabolic flows of Kahler potentials which arise naturally in this context. The first is the J-flow, which is the gradient flow of a functional appearing in Chen's formula for the Mabuchi energy. The study of the J-flow has led to significant advances in understanding the lower boundedness and asymptotics of the Mabuchi energy. The second is the Kahler-Ricci flow. Its behavior in the Fano case is not yet well understood, and it is proposed that the method of multiplier ideal sheaves may capture the necessary information about its singularities to be able to provide a link with stability. The third is the Calabi flow. It is a fourth order parabolic PDE about which little is known in general. The principal investigator intends to study the problem of long time existence of this flow. An important problem in geometry and physics is whether a given space has a special notion of distance. Take, for example, the two dimensional sphere - the surface of a ball. With our usual sense of distance, this space is curved in the same way at every point. We say that the sphere admits a 'metric of constant curvature'. Not all spaces admit such metrics, and it is an interesting and deep problem to find conditions under which they do. A natural and beautiful approach to this problem is the parabolic, or 'heat flow' method. The idea is simple. The distribution of heat in an object (not subject to outside sources) will flow in time, becoming more even and finally constant - no matter what the initial distribution looked like. In a similar way, if we start with an arbitrary notion of distance on a space, then we can apply a natural 'heat flow' and hope to prove, under the right conditions, that we obtain convergence to a metric of constant curvature, or some other special metric, as time evolves. If no such metrics exist, then we expect the flow to go wrong - to develop singularities. The PI intends to study the question of convergence and singularities of three such parabolic flows corresponding to different types of special metrics.