The Kahler Ricci flow and the extremal Kahler metrics

本课题主要研究Kaehler几何中的几个核心问题：极值Kaehlermetrics的唯一性和存在性，以及如何得到极值Kaehlermetrics的解。在KaehlerRicci流问题上，我和G. Tian一起工作。我们的主要结果是：在任何Kaehler-Einstein流形中，如果初始度规具有正的对分曲率，则流以指数收敛为唯一的Kaehler-Einstein度规。我打算和他一起消除用正对分曲率的假设或者凯勒-爱因斯坦度规的假设。在测地线问题上，我的主要研究结果是：a)在Kaehler类中任意两个Kaehler度量之间存在二阶导数一致有界的测地线。这个结果的一个直接推论是，如果流形的第一个chern类是负的，则常数标量曲率度规在每个Kaehler类中是唯一的。b) Kaehlermetrics空间是一个度量空间，在Alexandrov意义上是非正弯曲的。在测地线问题上，我想将测地线的规则性改进为三个导数一致有界（或者了解这种规则性何时可能失效）。这将是凯勒几何中一个非常重要的结论。凯勒·爱因斯坦度量法自然地产生于物理学、代数几何和其他不同的数学领域。极值凯勒度规是E. Calabi对这些概念的自然推广。求这些度量的方法是通过求解流形上的一个完全非线性椭圆型偏微分方程。大多数时候，人们无法明确地找到解决方案。那么我们就完全依赖于各种各样的先验估计来确定何时存在一个解，以及解度量是否应该是唯一的。在这个项目中，我们将开发一些新技术来处理这些困难的评估。这些技巧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.