The Kahler Ricci flow and the extremal Kahler metrics

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