The Kahler Ricci flow and the extremal Kahler metrics

Princeton University, Princeton NJ

Investigators

Abstract

Abstract for DMS - 0110321 This projects mainly deals with some central issues in Kaehler geometry: the uniqueness and existence of extremal Kaehler metrics and as well how to obtain such a solution. On the Kaehler Ricci flow problem, I am working with G. Tian. Our main results are: in any Kaehler-Einstein manifolds, if the initial metric has positive bisectional curvature, then the flow converges exponentially to a unique Kaehler-Einstein metric. I plan to work with him to eliminate with the assumption of positive bisectional curvature or the assumption of Kaehler Einstein metrics. On the problem of geodesic, the main results of my research are: a) there exists a geodesic with second derivatives uniformly bounded, between any two Kaehler metrics in a Kaehler class. A direct consequence of this result is that the constant scalar curvature metric is unique in each Kaehler class if the first Chern class of the manifold is negative. b) the space of Kaehler metrics is a metric space and it is non-positively curved in the sense of Alexandrov. On the problem of geodesic, I want to improve the regularity of geodesic to three derivatives uniformly bounded (or to understand when this regularity might fail). That will be a very important consequence in Kaehler geometry. Kaehler Einstein metric arose naturally from Physics, algebraic geometry and some other diverse areas of mathematics. Extremal Kaehler metric is a natural generalization of these concepts by E. Calabi. The method of finding these metrics is by solving a totally nonlinear elliptic partial differential equation on manifolds. Most of the time, one can not find solution explicitly. Then one has to rely on various kind a priori estimates to determined when there exists a solution and if the solution metric should be unique. In this project, we will develop some new techniques to handle these difficult estimates. And these techniques will have impact in other related problems of mathematics.