Mathematical Sciences: Differential Geometry and Partial Differential Equations

首席研究员Richard Schoen将继续研究满足曲率方程的黎曼度量的构造，以及涉及黎曼流形之间映射的微分方程。他将研究非紧流形的Yamabe型问题。对于紧流形，他将研究某些几何上的自然唯一性问题。对于调和映射，他将考虑平稳映射的正则性问题，能量泛函较高临界点的存在性问题，以及极小值的一般唯一性问题。他还打算研究一种几何上的自然泛函，这种泛函在用一个映射的逆替换它时是不变的。这应该是研究涉及一对一地图的情况的有用工具。具有一定曲率限制的度量的构造是微分几何的一个重要方面，在数学物理中具有分支。

The principal investigator, Richard Schoen, will continue his research on the construction of Riemannian metrics satisfying curvature equations, and on differential equations involving mappings between Riemannian manifolds. He will study the Yamabe- type problem for non-compact manifolds. For compact manifolds he will investigate certain geometrically natural uniqueness questions. For harmonic maps, he will consider regularity questions for stationary maps, existence questions for higher critical points of the energy functional, and generic uniqueness questions for minima. He also intends to study a geometrically natural functional which is unchanged upon replacement of a map by its inverse. This should be a useful tool for studying situations involving one-one maps. The construction of metrics with certain curvature restrictions is an important aspect of differential geometry with ramifications in mathematical physics.