Some Analytic and Geometric Problems of the Ricci Flow

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目旨在研究几何演化方程及其相关主题，特别是Ricci流。研究了Li-Yau-汉密尔顿型Harnack不等式，它们与Perelman？的熵泛函，以及它们在几何流中的应用。受汉密尔顿的影响？在奇点分析的基础上，我们研究了具有对称性的4-流形上Ricci流的奇点性质和古解。研究了Ricci流下各种几何算子的单调性。本文研究局部齐次流形上的几何流。这将有助于我们更好地理解几何流的长期行为。Ricci流一直是微分几何和拓扑学的研究热点。佩雷尔曼最近对汉密尔顿的研究？的Ricci流程序给出了长期存在的Poincare猜想的一个解决方案，也给出了几何化猜想的一个可能的解决方案。该项目将提高我们对Ricci流程序以及其他几何演化方程，如平均曲率流，高斯曲率流，交叉曲率流和重整化群流的理解。该项目还将提高我们对微分几何，抛物偏微分方程和数学物理等领域的理解。 教育部分将帮助研究生或其他领域的研究人员更好地理解利玛窦流动计划。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The project is aimed at the study of geometric evolution equations and related topics, especially the Ricci flow. We study Li-Yau-Hamilton type Harnack inequalities, their relations with Perelman?s entropy functional, and their applications to geometric flows. Motivated by Hamilton?s work on singularities analysis, we study properties of singularities and ancient solutions of the Ricci flow on 4-manifolds with symmetry. The monotonicity of various geometric operators under the Ricci flow is considered. We propose to investigate geometric flows on locally homogeneous manifolds. This will help us to understand the long time behavior of geometric flows better. The Ricci flow has been a center of attention of differential geometry and topology. Recent work of Perelman on Hamilton?s program of the Ricci flow gives a solution to the long-standing Poincare conjecture and also a possible solution to the geometrization conjecture. The proposed project will improve our understanding of the Ricci flow program and as well as other geometric evolution equations, such as mean curvature flow, Gauss curvature flow, cross curvature flow and renormalization group flow. The project will also enhance our understanding of other fields in differential geometry, parabolic PDE and mathematical physics. The education component will help graduate students or researchers in other fields to understand the Ricci flow program better.