曲率流中的若干理论、技巧及其应用研究

曲率流是流形上黎曼度量的发展方程，把初始度量演化为流形上允许的典则黎曼度量，是几何分析的中心之一。通过构造不同曲率流中的单调几何量，能够证明一些经典不等式。研究若干热类型方程的解，特别是对应的熵公式和Harnack不等式，能够得到流形的几何性质。.本项目计划发展若干有效理论和技巧，系统研究曲率流中熵的单调性及Harnack不等式，以及它们在Ricci流、平均曲率流、Gauss曲率流等的几何应用。我们的目标是：建立若干曲率流中的熵公式及其应用,包括类似于Perelman的W熵和其它类型的熵；建立若干新的梯度估计、Harnack不等式及其应用；利用熵的稳定性研究曲率流的特殊解，特别是Ricci孤立子、平均曲率流自收缩解等。我们预期在上述各方面取得进展，系统地发展曲率流研究中的理论、方法和技巧，丰富曲率流的研究手段。

A curvature flow on a manifold is the evolution equation of the metrics of one parameter, which evolves the initial metric to a canonical Riemannian metric of the manifold. It is now one of the central problems in geometric analysis. One can prove some classical inequalities by establishing monotone geometric quantities along different type curvature flows. In curvature flows, by studying solutions of heat type, in particular the corresponding entropy functionals and Harnack inequalities, one can get geometric information of the manifold. .In this project, by developing a few theories and techniques, we will study curvature flows in particular the Ricci flow, the mean curvature flow and the Gauss curvature flow. Our goals are to find new entropy functionals in geometric flows, including formulas of Perelman's type and other type; find new gradient estimates and Harnack inequalities and their applications to curvature flows; study special solutions of curvature flows via stability of entropy functionals, in particular the gradient Ricci soliton and self-similar shrinker of the mean curvature flow. We expect to make advances in the above mentioned aspects, and develope efficient theories, methods and techniques to study curvature flows.