Nonlinear geometric evolution equations

摘要：本课题主要研究两类问题。第一部分是里奇流方程的几何性质和解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一些非线性偏微分方程，PI得到的一些技术和结果是相关的，可以应用。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。这些结果对里奇流在流形几何拓扑结构研究中的应用具有深远的意义。本课题的第二个主题是li - you - hamilton型的尖锐梯度估计、相关单调性公式及其在几何非线性偏微分方程中的应用。它们在物理上的关系，特别是统计力学上的关系，也要加以研究。目的是发现一个基本的物理/几何原理来统一各种尖锐估计和单调性公式。为进一步发现其它几何偏微分方程的单调性新公式提供了指导。本课题研究数学和物理学中最重要的几何偏微分方程之一——里奇流的几何和解析性质。该项目的完成将加强目前对里奇方程奇点形成及其与应用科学和理论物理几个分支的关系的理解。该项目通过为包括高中生在内的普通观众举办讲座，撰写多部专著和调查文章，为提高数学界和公众的科学认识做出了贡献。

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.