Mathematical Sciences: Analytic Problems About Riemannian Metrics

首席研究员将继续研究 黎曼度量与谱几何。特别是，他将 考虑以下问题： 上给定数量曲率方程的可解性 球面，找到黎曼函数的Sobolev常数的界 流形的谱数据，并构造极值 关于谱泛函的度量 注意事项。 该奖项所支持的研究通常将试图 在流形或曲面上关联局部几何信息 关于表面的全局信息。当地的信息是 在表面上的偏微分方程， 全球信息涉及全球定义的解决方案， 偏微分方程将当地与 整体信息是微分几何中一个常见的信息。之一 这里要考虑的本地定义的运算符是 Laplacian可能是最重要的运算符， 应用于工程或物理。

The principal investigator will continue his study of Riemannian metrics and spectral geometry. In particular, he will consider the following problems: find conditions for the solvability of the equation of prescribed scalar curvature on spheres, find bounds for the Sobolev constant of a Riemannian manifold in terms of its spectral data, and construct extremal metrics with respect to functionals arising from spectral considerations. The research supported by this award generally will attempt to relate the local geometric information on a manifold or surface with global information about the surface. The local information is given in terms of partial differential equations on the surface and the global information involves globally defined solutions to these partial differential equations. The theme of relating local to global information is a common one in differential geometry. One of the locally defined operators to be considered here is the Laplacian which is perhaps the most important operator for applications to engineering or physics.