Mathematical Sciences: Differential Geometry and Partial Differential Equations

首席研究员理查德·舍恩将继续 他对黎曼度量构造的研究， 曲率方程，以及涉及 黎曼流形之间的映射 他将研究山边- 型问题的非紧流形。 对于紧致流形， 将研究某些几何学上的自然独特性 问题. 对于调和映射，他将考虑正则性 静态映射的问题，高阶映射的存在性问题 能量泛函的临界点和泛唯一性 最小值的问题 他还打算研究一个几何学 地图更换后不变的自然功能 其逆。 这应该是一个有用的工具， 涉及一对一映射的情况。 具有一定曲率的度量的构造 限制是微分几何的一个重要方面， 数学物理的分支

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.