Alexandrov空间上的Ricci曲率

本项目将深入研究Alexandrov空间上一个几何定义的Ricci曲率下界。Alexandrov空间是允许存在奇异集的几何对象。这个几何定义的Ricci曲率下界是由本项目申请者和朱熹平教授联合给出，并且我们已经得到一些它的几何与分析推论。本项目中，我们将主要应用分析手段去深入研究它的基本性质和所蕴含的几何推论。我们拟研究如下问题： （1）这个新定义的Ricci曲率下界是否等价于Sturm-Lott-Villani所定义的曲率-维数条件；（2）完备非紧Alexandrov空间上热方程的Li-Yau估计；（3）Alexandrov空间上Levy-Gromov等周不等式； （4）Alexandrov空间之间的调和映射的Lipschitz连续性问题。

This proposal is concerned with a geometric definition for lower Ricci curvature bounds on Alexandrov spaces. Alexandrov spaces are a class of geometric objects which allow the existence of singularities. This definition of lower Ricci curvature bounds is intruduced by the applicant and Prof. Xiping Zhu, and we have got some geometric and analyitic results under this Ricci condition.In this project we will emphasize on the analytic method in our research on both basic properties of this Ricci condition and its geometric and analytic consequences. We will study the following problems: (1) the equivalence of the new lower Ricci curvature bounds and Sturm-Lott-Villani's curvature-dimension condition; (2) Li-Yau estimates on complete non-compact Alexandrov spaces; (3) Levy-Gromov isoperimetric inequality on Alexandrov spaces; (4)the Lipschitz regularity of harmonic maps between Alexandrov spaces.