渐近平坦与渐近双曲流形中的几何分析问题

This project is mainly devoted to two types of geometric analysis problems on the conformally compact manifolds and asymptotically flat manifolds, specifically including: (1)Static solution to the Einstein-Maxwell-Scalar equations and its related geometric analysis problems; (2)The structure of conformal infinity for the conformally compact manifolds with scalar curvature satisfying certain conditions; (3) General Alexandrov-Fenchel (AF) type inequalities on the asymptotically hyperbolic and hyperbolic spaces; (4)Geometry and topology associated with the scalar curvature. These problems have a profound physical background and also are closely related to the frontier issues of geometric analysis. With scalar curvature as a link, they are intimately linked to each other. By investigating these problems, we strive not only to deepen the understanding of the geometric and topological properties for the manifolds under certain conditions on the scalar curvature, but also to promote the study of related physical problems. Furthermore, we hope to develop a series of approach to deal with the geometric problems on the manifolds with certain asymptotic structure at infinity.

本项目拟研究共形紧及渐近平坦流形上两类几何分析问题， 具体包括：Einstein-Maxwell-Scalar方程组静态解及相关的几何分析问题； 数量曲率满足一定条件的共形紧流形无穷远处结构问题；渐近双曲和双曲空间上的一般的Alexandrov-Fenchel（A-F）型不等式； 和数量曲率有关的几何拓扑。这些问题具有深刻的物理背景同时又和几何分析的前沿问题相联系。它们以数量曲率为纽带，互为表里，紧密地联系在一起。 通过研究这些问题，我们力求做到既加深对于数量曲率满足一定条件下的流形的几何拓扑性质的认识，又促进相关物理问题研究。同时，希望发展出一套处理无穷远处具有渐近结构流形上几何分析问题的方法。

本项目研究了共形紧及渐近平坦流形上两类几何分析问题， 具体包括： 数量曲率满足一定条件的共形紧流形无穷远处结构问题； 共形紧Einstein流形的刚性；渐近局部双曲流形的正能量问题； 和数量曲率有关的几何拓扑。这些问题具有深刻的物理背景同时又和几何分析的前沿问题相联系。它们以数量曲率为纽带，互为表里，紧密地联系在一起。 通过对这些问题的研究，加深了我们对于数量曲率满足一定条件下的流形的几何拓扑性质的认识，为数学广义相对论中的各种正能量型定理的研究提供了新的视角。