Partial regularity and rigidity problems associated to geometric elliptic systems

主要研究人员建议应用偏微分系统中的技巧和方法来研究微分几何中的正则性和刚性问题。第一个提出的问题是关于某些渐近平坦流形的刚性。这个项目是研究人员早期关于度量的渐近衰减的工作的继续，当时她在椭圆系统中应用了一种分析方法。其中的椭圆系是生物和化学中常见的反应扩散型系统。刚性问题可以看作是度量的渐近衰减问题的一个极端情况。第二个项目是关于几何椭圆系统在L^2范数界下的部分正则性。这个问题在模空间的研究中自然而然地出现了。模空间的概念是描述黎曼度量的拓扑结构和分析结构的一种现代进步。研究人员建议用以前在调和映射和杨-米尔斯方程中使用的类似分析方法来研究正则性理论。该研究包含了微分几何和应用数学之间的交叉研究。在几何方面，这两个问题都是在总体上理解黎曼度量结构的更大计划内的。为了描述这种结构，开发一种测量空间粗糙度的工具--局部规则性是必不可少的。从分析的角度来看，这些问题被证明是一个(静态)反应-扩散系统，这是某些科学领域中的一种典型系统。在几何学的部分正则性文献中，很少有人知道这一方向上的联系(调和映射和杨-米尔斯方程是少数几个)。这位研究人员计划致力于这一方向，并将她通过拟议的活动获得的知识传播给不同的几何学家和应用数学家。

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.