Nonlinear geometric evolution equations

DMS-0805143本项目关注的问题有两类。第一部分是关于Ricci流方程的几何性质和分析性质及其在流形几何和拓扑研究中的应用。其次，在应用科学中出现了几个非线性偏微分方程组，PI所得到的一些技巧和结果是相关的和可以应用的。项目的第一部分包括自相似解的分类，即梯度孤子，以及高维Ricci流解的凸型估计。这些结果对Ricci流在流形几何拓扑结构研究中的应用有着深远的影响。本项目的第二个主题是Li-Yau-Hamilton型的尖锐梯度估计，相关的单调性公式及其在几何非线性偏微分方程组中的应用。它们与物理学的关系，特别是统计力学，也应加以研究。其目的是发现一个基本的物理/几何原理，以统一各种精确的估计和单调公式。它还将为进一步发现其他几何偏微分方程中新的单调性公式提供指导。该项目是关于数学和物理中最重要的几何偏微分方程之一的Ricci流的几何和分析性质。该项目的完成将加深对Ricci方程奇点形成及其与应用科学和理论物理几个分支之间关系的当前理解。该项目通过为包括高中生在内的广大听众讲课，撰写几部专著和调查文章，有助于传播增进对数学界和普通公众的科学理解。

Abstract - DMS - 0805143There are two classes of problems on which this project isfocused. The first is about the geometric and analyticproperties of the Ricci flow equation and their applications tothe study of geometry and the topology of manifolds. Secondly,there are a couple of nonlinear partial differential equationsappeared in the applied science, to which some techniques andresults obtained by the PI are related and can be applied. Thefirst part of project consists of the classifications ofself-similar solutions, namely gradient solitons, and theconvexity type estimates for Ricci flow solution in highdimensions. Such results have far-reaching consequences in theapplications of Ricci flow in the study ofgeometric-topological structure of the manifolds. The secondtheme of the project is about the sharp gradient estimates ofLi-Yau-Hamilton type, related monotonicity formulae and theirapplications in geometric nonlinear PDEs. Their relations tophysics, statistical mechanics in particular, shall be studiedtoo. The aim is to discover a fundamental physical/geometricprinciple to unify various sharp estimates and monotonicityformulae. It will also provide the guideline for furtherdiscovery of the new monotonicity formulae in other geometricpartial differential equations.The project is on the geometric and analytic properties ofRicci flow, one of the most important geometric partialdifferential equation in mathematics and physics. Thecompletion of the project will enhance the currentunderstanding of the singularity formation of the Ricci flowequation and its relation with several branches of appliedscience and theoretic physics. By giving lectures for generalaudience including high school students, writing severalmonographs and survey articles, the project contributes to thedissemination of enhancing the scientific understanding ofmathematical community as well as the general publics.