Linear and nonlinear geometric evolution equations

线性和非线性几何演化方程

• 批准号: 1401500
• 负责人: Lei Ni
• 金额: $ 16.67万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401500&HistoricalAwards=false
• 关键词: Linear; nonlinear; geometric; evolution; equations

Differential geometry studies the topological, geometric and analytic properties of the spaces. Since all physical events take places in a space, by Einstein's relativity the spatial metric properties of the underlying space have fundamental impacts on the physical events happening in the space, including the very one we live in. This relation places the subject of the differential geometry in a central position of the mathematics and some fundamental issues of sciences. Modern differential geometry appeals to the theory/methods of linear and nonlinear partial differential equations. The proposal involves the study of differential geometry via the method of the evolutional partial differential equations with the focus on the study of the entropy, the monotonicity, and related sharp differential estimates motives by thermodynamics and statistical mechanics. The study of Gauss curvature flow has important consequences on the subjects of the image processing, affine differential geometry and convex geometry. The study of the Ricci flow in the proposal advances the understanding of the differential topological structure of the spaces and is related to the theoretic physics with direct bearing on the high energy physics. The resolving of the problems involved in the proposal advances the above mentioned related subjects in sciences. The out-reach components of the proposal disseminate the new results to general public and contributes towards the mathematical education in the community in the greater area of San Diego and beyond. It also contributes to the training and the early career development of under-represented groups. The technical aspects of the proposal involve the development of the sharp monotonicity of the entropy and related sharp point wise estimates. The study of differential geometric problems relies on solving evolution equations such as the Ricci flow equation, Gauss curvature flow equation or other nonlinear evolution equations, and/or the study of delicate properties of the solutions obtained. The theory of modern partial differential equations reduces the study of the solutions of various linear and nonlinear equations to the monotonicity of certain entropic quantities and related point wise estimates. The de-regularizing estimates proposed in this proposal broadens the scope of the already far-reaching gradient estimates, Hessian estimates and curvature estimates techniques developed in the last several decades to more general settings, by requiring less regularity of the solutions involved, while encoding far more geometric information. They have produced and will produce much powerful geometric consequences. The techniques developed can in turn advances the study of the related partial differential equations, which usually occupies a central role in the theory of the partial differential equations.

微分几何研究空间的拓扑、几何和解析性质。由于所有物理事件都发生在一个空间中，根据爱因斯坦的相对论，底层空间的空间度规属性对发生在空间中的物理事件有根本影响，包括我们生活的那个空间。这种关系使微分几何这门学科在数学和一些基本科学问题中处于中心位置。现代微分几何要求线性和非线性偏微分方程解的理论和方法。该方案涉及用演化偏微分方程组的方法研究微分几何，重点是用热力学和统计力学研究熵、单调性和相关的尖锐微分估计动机。高斯曲率流的研究在图像处理、仿射微分几何和凸几何等学科中具有重要的意义。文中对Ricci流的研究，加深了对空间微分拓扑结构的理解，与高能物理有着直接的理论联系。提案中涉及的问题的解决，推进了上述科学相关学科的发展。该提案的外展部分向公众传播新的成果，并为圣地亚哥及其他地区社区的数学教育作出贡献。它还有助于对任职人数不足的群体进行培训和早期职业发展。该方案的技术方面涉及发展信息熵的尖锐单调性和相关的尖锐的逐点估计。微分几何问题的研究依赖于求解发展方程，如Ricci流动方程、Gauss曲率流动方程或其他非线性发展方程，和/或研究所获得的解的精细性质。现代偏微分方程组理论将对各种线性和非线性方程的解的研究归结为某些熵的单调性和相关的逐点估计。该建议中提出的去正则化估计将过去几十年开发的已经影响深远的梯度估计、黑森估计和曲率估计技术的范围扩大到更一般的设置，因为所涉及的解的规则性较低，而编码的几何信息要多得多。它们已经产生并将产生更强大的几何后果。所发展的技术反过来可以促进对相关偏微分方程组的研究，而偏微分方程组通常在偏微分方程组理论中占据核心地位。