Ricci曲率有界流形的几何结构及相关应用

The goal of the project is to study various geometric curvature equations in Riemannian manifolds and their applications. The project investigator will mostly center around the study of geometric structures of manifolds with bounded Ricci curvature, and apply it to Ricci flow and Bergman kernel estimates. This project would depend on Cheeger, Colding, Naber and Tian’s singularity theory in limit spaces, and methods developed in geometric analysis and geometric measure theory. ..There are three research topics in this project. The first topic of the project centers on studying finite diffeomorphism of manifolds with bounded Ricci curvature. The main idea is to decompose the manifolds and control the intersection domains of the decomposition. The second topic mainly concerns the extension and convergence of compact Ricci flow with bounded scalar curvature. We will focus on the study on distance functions and the structure of the limit spaces. The last topic is about best lower bound estimate for Bergman kernels on Fano manifolds with bounded Ricci curvature. Each of these topics somehow concerned with the intrinsic structure of higher dimensional geometries. Each of these topics largely depends on the structure of manifolds with bounded Ricci curvature. The project will address some basic problems of these topics.

本项目旨在研究黎曼流形的几何曲率方程及其应用。申请人将利用几何分析和几何测度论中的方法，在Cheeger, Colding, Naber和田刚等人关于极限空间的奇点理论基础上，深入研究Ricci曲率有界流形的几何结构，并应用到Ricc流和Bergman核估计等问题。.. 本项目包含三个研究课题。第一个课题是考虑Ricci曲率有界的黎曼流形的微分同胚有限性，主要是对Ricci曲率有界流形的分解与分解相交区域的研究。第二个课题是研究数量曲率有界的紧Ricci流的延拓与收敛性，主要研究Ricci流的距离函数与极限奇点的结构。 第三个课题是研究Ricci曲率有界的Fano流形的Bergman核的最优下界估计。每个课题都与高维流形的内蕴几何结构有极大的关联，都依赖Ricci曲率有界流形的几何结构的研究。本项目将会解决这些课题的部分基本问题。

本项目主要研究里次曲率极限空间的结构，证明爱因斯坦流形的曲率L^2积分猜想，爱因斯坦流形极限的奇点有限测度猜想。该项目的另一个成果是关于里次曲率有下界流形的极限空间的结构，证明了极限奇点集的结构定理。