Mathematical Sciences: Analytic Problems About Riemannian Metrics

首席研究员将继续他的黎曼度量和光谱几何的研究。他将特别考虑以下问题：寻找球面上规定标量曲率方程的可解性的条件，根据谱数据找到黎曼流形的Sobolev常数的界，并构造关于由谱考虑引起的泛函的极值度量。该奖项支持的研究通常将尝试将流形或曲面上的局部几何信息与有关该曲面的全局信息联系起来。局部信息以曲面上的偏微分方程的形式给出，全局信息涉及这些偏微分方程的全局定义解。将局部信息与全局信息联系起来是微分几何中一个常见的主题。这里要考虑的一个局部定义算子是拉普拉斯算子，它可能是工程或物理应用中最重要的算子。

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.