Partial regularity and rigidity problems associated to geometric elliptic systems

与几何椭圆系统相关的部分正则性和刚性问题

• 批准号: 1104592
• 负责人: Fernando Marques
• 金额: $ 16.19万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104592&HistoricalAwards=false
• 关键词: Partial; regularity; rigidity; problems; associated

The principal investigator proposes to apply techniques and methods in partial differential systems to study regularity and rigidity problems in differential geometry. The first proposed problem is on the rigidity of certain asymptotically flat manifolds. This project is a continuation of the investigator's earlier work on asymptotic decay of metrics where she applied an analysis method in elliptic systems. The elliptic systems therein are of reaction-diffusion type, which appears often in biology and chemistry. The rigidity problem can be viewed as an extremal case of the asymptotic decay problem of metrics. The second proposed project is on partial regularity of geometric elliptic systems under L^2 norm bound of curvatures. Such problem arises naturally in the study of moduli spaces. The notion of moduli spaces is a modern advance in describing the topological and analytical structure of Riemannian metrics. The investigator proposes to study the regularity theory by a similar analysis approach previously used in harmonic maps and Yang-Mills equations.The proposed research contains an interdisciplinary study among differential geometry and applied mathematics through partial differential equations. On the geometrical side, both problems are within a larger program in understanding the structure of Riemannian metrics on the whole. In order to describe the structure, it is essential to develop a tool, partial regularity, to measure the roughness of the space. From analytical point of view, the problems turn out to be characterized by a (static) reaction-diffusion system, a typical type of systems in some areas of sciences. In the literature of partial regularity in geometry, there were few connections known in this direction ( harmonic maps and Yang-Mills equations are among the few). The investigator plans to devote herself in this direction and disseminate the knowledge she obtained through the proposed activity among both differential geometers and applied mathematicians.

主要研究者建议在部分差分系统中应用技术和方法，以研究差异几何形状中的规律性和刚性问题。 第一个提出的问题是关于某些渐近平坦的歧管的刚度。该项目延续了研究者在渐近衰变指标上的早期工作，她在椭圆系统中应用了分析方法。其中的椭圆系统具有反应扩散类型，通常出现在生物学和化学中。刚性问题可以看作是指标的渐近衰减问题的极端情况。第二个提议的项目是基于曲线的l^2规范结合的几何椭圆系统的部分规律性。这种问题自然出现在模量空间的研究中。模量空间的概念是描述里曼尼亚指标的拓扑结构和分析结构的现代进步。研究者建议通过先前在谐波图和杨麦尔式方程中使用的类似分析方法来研究规则性理论。拟议的研究包含差异几何形状和通过部分微分方程应用数学之间的跨学科研究。从几何方面来说，这两个问题都在一个更大的程序中，在理解总体而言的里曼尼亚指标的结构方面。为了描述结构，必须开发一种工具，部分规律性，以测量空间的粗糙度。从分析的角度来看，这些问题的特征是（静态）反应扩散系统，这是科学某些领域的典型系统类型。在几何学部分规律性的文献中，在这个方向上很少有连接（谐波图和杨米尔斯方程是少数几个）。研究人员计划朝这个方向朝着这个方向发展，并传播她通过差异几何学和应用数学家之间提议的活动获得的知识。