The Kahler Ricci flow and the extremal Kahler metrics

DMS -0110321的摘要本项目主要研究Kaehler几何中的一些中心问题：极值Kaehler度量的唯一性和存在性以及如何获得这样的解。在KaehlerRicci流问题上，我与G.田我们的主要成果是：在任何Kaehler-Einstein流形中，如果初始度量具有正的对分曲率，则流指数收敛到唯一的Kaehler-Einstein度量。 我计划和他一起工作，用正二分曲率的假设或凯勒爱因斯坦度量的假设来消除。在测地线问题上，我们的主要研究结果是：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，常数标量曲率度量是唯一的，在每个Kaehler类，如果第一陈类的流形是负的。B）Kaehler度量空间是度量空间，并且是Alexandrov意义下的非正曲空间。在测地线的问题上，我想把测地线的正则性改进为三阶导数一致有界（或者理解这种正则性何时失效）。这将是Kaehler几何中一个非常重要的推论。Kaehler Einstein度规自然地产生于物理学、代数几何和其他一些不同的数学领域。极值Kaehler度量是E.卡拉比 找到这些度量的方法是通过求解流形上的完全非线性椭圆偏微分方程。大多数时候，人们无法明确地找到解决方案。那么就必须依赖于各种先验估计来确定解的存在性以及解的度量是否唯一。 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。这些技术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.