Mathematical Sciences: Analytic Problems About Riemannian Metrics

首席研究员将继续他对黎曼度量和谱几何的研究。特别是，他将考虑以下问题：找到球面上规定的标量曲率方程可解的条件，根据黎曼流形的谱数据找到其Sobolev常数的界，并构造关于谱考虑产生的泛函的极值度量。该奖项支持的研究通常会尝试将流形或曲面上的局部几何信息与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组的形式给出，而全局信息涉及这些偏微分方程解的整体定义。在微分几何中，将局部信息与全局信息联系起来是一个常见的主题。这里要考虑的局部定义的算符之一是拉普拉斯算符，它可能是工程或物理应用中最重要的算符。

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.