Analytic and Geometric Aspects of Ricci Flow

摘要DMS - 0203926。本项目的目的是研究黎曼度量的哈密顿里奇流的解析和几何方面，以及相关主题。特别地，首席研究员建议研究Ricci流解的注入半径估计。PI提出对Ricci流和线性化Ricci流研究li - yu - hamilton （Harnack）不等式。PI还提出研究Ricci流解的坍缩序列及其紧性理论。最后，PI提出研究线性化Ricci流的估计及其应用。PI将侧重于维度3，在这里可用的工具最多，所提出的问题具有最多的拓扑应用。里奇流是一种使曲面和流形上的几何结构发生变形的演化方程，它既能使结构得到改善，又能使结构变得平滑。它是近年来几何演化方程快速发展的前沿，对平均曲率流和许多其他几何演化方程的研究产生了重大影响。对流动下奇点的理解工作导致了新的、强大的分析和几何工具的发展，这些工具在许多几何演化方程的研究中得到了应用。许多现象都是用几何演化方程来模拟的，比如热传递、熔融金属和固体金属之间的界面演化以及冰和水之间的界面。几何演化方程也用于平滑图像，并在计算机视觉中有应用。

ABSTRACT DMS - 0203926.The objective of this project is to study analytic and geometric aspects of Hamilton's Ricci flow of Riemannian metrics, and related topics. In particular, the principal investigator proposes to investigate injectivity radius estimates for solutions to the Ricci flow. The PI proposes to studyLi-Yau-Hamilton (Harnack) inequalities for both the Ricci flow and the linearized Ricci flow. The PI also proposes to study collapsing sequences of solutions to the Ricci flow and the related compactness theory. Finally, the PI proposes to study estimates for the linearized Ricci flow andtheir applications. The PI will focus on dimension 3, where there are the most tools available and the problems posed have the most topological applications.The Ricci flow is an evolution equation which deforms geometric structures on surfaces and manifolds both improving and smoothing out the structures. It has been at the cutting edge of the recent rapid progress in geometric evolution equations and has greatly influenced the study of the mean curvatureflow and many other geometric evolution equations. The work on understanding the singularities which develop under the flow has led to the development of new and powerful analytic and geometric tools, which have found applications in thestudy of many geometric evolution equations. Many phenomena aremodeled by geometric evolution equations, such as heat transfer, the evolution of interfaces between molten and solid metal, and the interfaces between ice and water. Geometric evolution equations are also used to smooth out images and have applications in computer vision.