The Kahler Ricci flow and the extremal Kahler metrics

对于DMS-0110321，这个项目主要研究Kaehler几何中的一些核心问题：极值Kaehler度规的唯一性和存在性，以及如何获得这样的解。在KaehlerRicci流动问题上，我与G.Tian一起工作。我们的主要结果是：在任何Kaehler-Einstein流形中，如果初始度量具有正对分曲率，则流指数收敛到唯一的Kaehler-Einstein度量。我计划与他合作，消除正对分曲率假设或凯勒·爱因斯坦度规的假设。关于测地线问题，我的主要研究结果是：a)在Kaehler类中的任意两个Kaehler度量之间存在一条二阶导数一致有界的测地线。这个结果的一个直接结果是，如果流形的FirstChern类是负的，则常量曲率度量在每个Kaehler类中是唯一的。B)Kaehler度量空间是度量空间，在Alexandrov意义下是非正曲线的。关于测地线问题，我想把测地线的规则性改进为三个导数一致有界(或者理解这种规则性何时可能失效)。这将是凯勒几何学中一个非常重要的结果。凯勒·爱因斯坦度规自然而然地起源于物理学、代数几何和其他一些不同的数学领域。极值Kaehler度量是E.Calabi对这些概念的自然推广。求这些对称性的方法是在流形上解一个完全非线性的椭圆型偏微分方程。在大多数情况下，人们无法明确地找到解决方案。然后，人们不得不依靠各种先验估计来确定何时存在解，以及解的度量是否应该是唯一的。在这个项目中，我们将开发一些新的技术来处理这些困难的估计。这些技巧还将对其他相关的数学问题产生影响。

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.