Mathematical Sciences: Submanifold Geometry and Integrable Systems

9626130 Terng提出的研究方向是微分几何中的子流形理论。研究者计划研究具有简单焦点结构的完全黎曼流形的子流形。特别是，她对紧黎曼流形中具有恒定焦半径的子流形和完全黎曼流形中的紧浸没感兴趣。此外，研究者还将与K. Uhlenbeck合作开展一个关于可积哈密顿系统及其在谐波映射和等距浸入中的应用的项目。子流形理论是几何学中一个庞大而古老的领域之一。最初的动机来自于空间中的表面，在本世纪之前，在构造满足某些特殊性质的表面上做了大量的工作，其中许多与曲率有关。提出的研究旨在利用现代技术深入研究其中的两种特殊性质：具有简单焦点结构的表面，以及允许“Bianchi-Backlund”转换的表面。

9626130 Terng The proposed research lies in the area of submanifold theory in differential geometry. The investigator plans to study submanifolds of complete Riemannian manifolds that have simple focal structures. In particular, she is interested in examining submanifolds with constant focal radii in compact Riemannian manifolds and taut immersions into complete Riemannian manifolds. In addition, the investigator is to engage in a collaborative project with K. Uhlenbeck on integrable Hamiltonian systems and their applications to harmonic maps and isometric immersions. Submanifold theory is a vast and one of the oldest areas in geometry. Original motivation comes from surfaces in space, where a great deal of work was done prior to this century on constructing surfaces satisfying certain special properties, many of them having to do with curvature. The proposed research aims to delve into two of these special properties using modern techniques: surfaces with a simple focal structure, and surfaces admitting 'Bianchi-Backlund' transformations.