Parabolic differential equations and the geometry of manifolds

抛物型微分方程和流形几何

• 批准号: 0805834
• 负责人: Brett Kotschwar
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805834&HistoricalAwards=false
• 关键词: Parabolic; differential; equations; geometry; manifolds

The proposed research concerns several problems in the study of nonlinear parabolic PDE and their application to the geometry and topology of manifolds. One general area of concentration will be that of singularity formation in the Ricci flow, with an emphasis on the study of ancient solutions and Ricci solitons. A central theme in the theory developed by Hamilton, Perelman, and others, is that the local geometry of a developing singularity possesses a relatively rigid structure, modelled, in many important cases, upon these special types of solutions. It is thus desirable to have as detailed a knowledge as possible of the diversity of forms which they can assume.Additionally, the co-PI proposes to investigate Type-II singularity development and the question of unique continuation for the Ricci flow equation. Another general area of concentration will be curvature flows of hypersurfaces, with an emphasis on the application, interpretation, and further development of differential Harnack inequalities for these flows. One aim will be to refine such inequalities in the setting of evolving spacelike hypersurfaces in Minkowski space, with an eye toward the study of eternal solutions and translating solitons. In the case of the mean curvature flow, the latter objects are of interest to researchers in general relativity as natural foliations of Lorentzian spacetimes. In another direction, the co-PI proposes to pursue a potential connection from this setting to the cross-curvature flow, an intrinsic flow of potential use in the study of three-manifolds with negative curvature.The Ricci flow and other geometric evolution equations considered in this proposal are representatives of the "heat-flow" method in geometry, the techniques and objectives of which straddle the field's lively interface with topology, analysis, and mathematical physics.This method has proven effective in attacking certain cases of one of the most fundamental questions in mathematics, namely, which manifolds admit constant curvature or otherwise canonical geometries? Not only does this question have ramifications for physical models of the universe, but the development of tools attendant to the approach promise to pay continued dividends to the analysis of the many structurally similar nonlinear PDE which occur as models of diverse phenomena throughout the physical sciences.

本论文主要研究非线性抛物型偏微分方程及其在流形几何和拓扑中的应用。一个集中的一般领域将是奇性形成的里奇流，重点是古代的解决方案和里奇孤子的研究。汉密尔顿、佩雷尔曼和其他人发展的理论的一个中心主题是，发展中奇点的局部几何具有一个相对刚性的结构，在许多重要情况下，以这些特殊类型的解为模型。 因此，希望有尽可能详细的知识的多样性的形式，他们可以assumption.Additionally，共同PI建议调查II型奇点的发展和问题的唯一延续的里奇流方程。 另一个集中的领域是超曲面的曲率流，重点是这些流的微分Harnack不等式的应用，解释和进一步发展。一个目标是在闵可夫斯基空间中的演化类空超曲面的背景下完善这种不等式，着眼于永恒解和平移孤子的研究。 在平均曲率流的情况下，后者的对象是广义相对论的研究者感兴趣的洛伦兹时空的自然叶理。 在另一个方向上，co-PI提出从这个设置到交叉曲率流的潜在联系，交叉曲率流是一种潜在的内在流，用于研究具有负曲率的三流形。在这个建议中考虑的Ricci流和其他几何演化方程是几何学中“热流”方法的代表，其技术和目标跨越了该领域与拓扑学的生动界面，这种方法已被证明是有效的攻击某些情况下的一个最基本的问题在数学上，即，哪些流形承认常曲率或以其他方式规范的几何？这个问题不仅对宇宙的物理模型有影响，而且伴随着这种方法的工具的发展，有望为分析许多结构相似的非线性偏微分方程提供持续的红利，这些偏微分方程作为整个物理科学中各种现象的模型而出现。