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.
