Analytic and Geometric Aspects of Ricci Flow

里奇流的解析和几何方面

• 批准号: 0203926
• 负责人: Bennet Chow
• 金额: $ 22.8万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203926&HistoricalAwards=false
• 关键词: Analytic; Geometric; Aspects; Ricci; Flow

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.

摘要DMS -0203926.本计画的目标是研究黎曼度规的汉密尔顿里奇流的解析与几何性质，以及相关的主题。特别是，首席研究员建议调查的Ricci流的解决方案的注入半径估计。PI提出研究Ricci流和线性化Ricci流的Li-Yau-汉密尔顿（Harnack）不等式。PI还建议研究Ricci流和相关紧性理论的解的塌陷序列。最后，PI提出研究线性化Ricci流的估计及其应用。PI将专注于3维，其中有最多的工具可用，所提出的问题具有最多的拓扑应用。Ricci流是一个演化方程，它使曲面和流形上的几何结构变形，改善和平滑结构。它是近年来几何发展方程研究的前沿，对平均曲率流等几何发展方程的研究产生了重要影响。对流动下发展的奇点的理解工作导致了新的和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。许多现象都可以用几何演化方程来模拟，如传热、熔融金属与固体之间的界面演化、冰与水之间的界面演化等。几何演化方程也用于平滑图像，并在计算机视觉中有应用。