应力-能量张量是研究能量泛函临界点能量行为的重要工具，在众多几何分析问题中有着重要的应用。我们将利用应力-能量张量在局部共形平坦流形和weighted 流形上建立相关几何量的单调不等式，然后在增长性条件下得到消灭定理并给出几何应用。通过计算某些几何量的Laplacian可以得到一类微分不等式，我们将研究其解的消灭定理并应用到具体的几何模型中得到刚性定理。本项目的另一个主要方面是研究黎曼流形的一些整体刚性现象，包括f-极小子流形的Bernstein型定理和共形平坦流形的整体刚性问题，同时我们还将研究常weighted平均曲率子流形，特别是f-极小子流形的曲率与几何拓扑结构的关系，以及无穷远处的拓扑结构。

Stress-energy tensor is a useful tool for investigating the energy behaviour of the critical point of energy functional, and also has important applications in geometric analysis. In this project, we will use the stress-energy tensors to establish monotonicity formulae for the relevant geometric quantities on locally conformally flat manifolds and weighted manifolds, and then deduce vanishing theorems under growth conditions and will give their geometric applications. By caculating the laplacian of some geometric quantities, we can get a differential inequality. We will investigate the vanishing theorems for soluitions of such differential inequality and give their geometric applications in specific geometry model. Another major object of the project is the global rigidity phenomenon of Riemannian manifolds, which includes Bernstein type theorem for submanifolds and global rigidity theorem for conformally flat manifolds. Furthermore, we will study the relationship between curvature and the geometric and topological structure of subamnifolds with constant weighted mean curvature, especially for f-minimal submanifolds, and their topological structure at infinity.