现代微分几何中的中心问题之一就是给定流形上典型度量的存在性问题。我们利用热流方法称为yamabe流方法来研究流形上常数曲率度量的存在性问题；其目的就是用先验估计和爆破分析手法把一个度量用这个热流形变成常数曲率度量。我们主要研究(完备非紧流形上的)yamabe流和相关问题，比如空隙定理；对于非紧的黎曼流形，yamabe流问题的研究相对比较少；在1998年我们研究了这个流并利用和发展了极值原理方法（或bernstein型估计方法）。 在本项目里，我们也研究Hamilton做出的如下猜想：对于一个维数大于2的紧的黎曼流形，对于任何初始度量的yamabe流，它在时间无穷处是收敛到一个常数曲率的度量。相关其他几何演化方程也会考虑。

One of the central problems in modern differential geometry is the existence of canonical metrics on a given manifold. We use heat flow method called the Yamabe flow to study the existence of metrics with constant scalar curvatures. The goal is to deform a given metric to a metric with constant scalar curvature via an apriori estimate and blowup analysis. We mainly concern with the Yamabe flows (on complete noncompact Riemannian manifolds) and related problems, such as the gap theorems. It is relatively few results about the Yamabe flow on noncompact manifolds. We have studied this flow in 1998 and we have developed the maximum principle method (or The Bernstein method). In this project, we also consider the Hamilton conjecture that the yamabe flow on a compact manifold converges to a metric with constant scalar curvature. Related geometric evolution equations will be studied.