一个黎曼度量称为临界度量，如果它的Ricci曲率张量满足某种Bochner型公式，其中包括爱因斯坦度量，具有常数量曲率的局部共形平坦度量。Ricci孤立子在weighted意义下也可以看作具有临界度量。在本项目，我们将首先利用应力-能量张量在光滑度量测度空间和局部共形平坦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。同时我们将通过计算某些几何量的Laplacian得到一类微分不等式，然后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，同时还将利用广义极大值原理来研究点点pinching问题。本项目的另一个主要方面是研究Ricci孤立子的几何结构，以及整体刚性问题。最后将研究与上述几何模型相关的子流形的Bernstein型定理，以及Ricci孤立子和共形平坦流形上调和p-形式的消灭定理。

A Riemannian metric is said to be critical if its Ricci curvature tensor satisfies a Bochner type formula, which include Einstein metrics, locally conformally flat metrics with constant scalar curvature. Ricci solitons also can be seen as a critical metrics in the weighted sence. In this project, we will first use the stress-energy tensors to establish monotonicity formulae for the relevant geometric quantities on smooth metric measure space and locally conformally flat manifolds, and then deduce vanishing theorems under growth conditions, and furthermore, give their geometric applications. At the same time, by caculating the Laplacian of some geometric quantities, we can get a general differential inequality. We will investigate the vanishing theorems for soluitions of such inequality and give applications on some specific models, and study pointwise pinching problem by generalize maximum principle. Another major object of the project is the geometric structure and the global rigidity phenomenon of Ricci solitons. Finally, we will study the Bernstein type problem of submanifolds which have critical metrics. and the vanishing theorems of harmonic p-forms on Ricci solitons and conformally flat manifolds.