探索紧致带边流形上一致退化椭圆型几何方程解的边界行为与流形本身的整体几何拓扑性质之间的关系是几何分析与偏微分方程中的热点研究课题之一，具有重要意义。本项目将以研究奇异 Yamabe 度量的边界展开的整体项的性质和 Ricci 曲率的负性为主要研究目标展开这方面的研究。具体来说，内容包括：. 1.研究奇异Yamabe度量的边界展开中的整体项的性质。目标是利用整体项的系数在流形边界上的积分给出流形本身的几何拓扑信息。申请人在这方面已有部分进展，完成两篇[46，47]。. 2.研究奇异Yamabe度量的Ricci曲率的负性（即Ricci曲率是否在整个流形上为负）。我们主要集中于两方面的研究：（1）给出 Ricci曲率的负性成立的一些几何拓扑条件；（2）给出 Ricci 负性不成立的紧致带边流形的一般性构造。申请人在这方面已有部分进展，完成1篇文章[21]。

Exploring the relationship between the boundary behavior of solutions of uniformly degenerate elliptic geometric equations on compact manifolds with boundary and the global geometric and topological properties of manifolds is a hot topic of importance in geometric analysis and partial differential equations. The main purpose of this project is to study properties of the global term in the boundary expansion of the singular Yamabe metrics and the negativity of Ricci curvature of the singular Yamabe metrics. In particular, the project can be divided into two parts:.1. Study properties of the integral term in the boundary expansion of the singular Yamabe metric. We expect to give the geometric and topological information of the manifold by studying the integral of the global coefficient on the boundary of the manifold. Some partial results have already been obtained by the applicant in [46,47]..2. Study the negativity of the Ricci curvature of the singular Yamabe metric (i.e. whether the Ricci curvature can be negative over the entire manifold). We mainly study from two aspects: (1) We aim to provide some geometric and topological conditions for the negativity of the Ricci curvatures of the singular Yamabe metrics to hold. (2) Give general construction of compact manifolds with boundary where the negativity of the Ricci curvature does not hold. Some partial results have already been obtained by the applicant in [21].