Partial regularity and rigidity problems associated to geometric elliptic systems

主要研究者提出应用偏微分系统的技术和方法来研究微分几何中的正则性和刚性问题。第一个问题是关于一类渐近平面流形的刚性问题。该项目是研究者早期关于度量渐近衰减的工作的延续，她在椭圆系统中应用了一种分析方法。其中椭圆型体系为反应扩散型体系，在生物和化学中经常出现。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。第二个课题是关于几何椭圆系统在曲率的L^2范数界下的部分正则性。这样的问题在模空间的研究中自然出现。模空间的概念是描述黎曼度量的拓扑结构和解析结构的一个现代进步。研究者建议用先前用于谐波映射和Yang-Mills方程的类似分析方法来研究正则性理论。本研究以偏微分方程为基础，将微分几何与应用数学相结合。在几何方面，这两个问题都在一个更大的程序中，在整体上理解黎曼度量的结构。为了描述结构，必须开发一种工具，部分规则，以测量空间的粗糙度。从分析的角度来看，问题的特征是（静态）反应扩散系统，这是某些科学领域的典型系统类型。在关于几何部分正则性的文献中，在这个方向上很少有已知的联系（调和映射和杨-米尔斯方程是其中的少数）。研究者计划致力于这一方向，并在微分几何学者和应用数学家之间传播她通过拟议活动获得的知识。

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.