Differential Geometry by way of Partial Differential Equations

抽象DMS-0202508第一个被提议的项目是理解完备流形的拓扑结构，这些完备流形的Ricci曲率由拉普拉斯算子的谱的底部的负倍数从下有界。在这种情况下，假设频谱的底部为正。第二个项目是研究具有非负截面曲率的完备流形上具有多项式增长的调和函数空间的大小。特别地，主要目的是证明次数不超过d的多项式增长的调和函数空间的维度被相同维度的欧氏空间中的类似空间的维度所有界。第三个项目是了解欧氏空间中稳定的极小超曲面的几何。更一般地，他想研究具有有限指标的极小超曲面。国际和平研究所还建议支持博士后孙晓峰博士的研究项目。孙博士建议将调和映射的正则性理论研究到度量空间。这些拟议的项目有一个统一的主题，即使用分析技术来研究下划线的几何空间。在许多情况下，通过对定义在空间上的适当偏微分方程解的详细分析，可以得出拓扑和几何结论。

ABSTRACT DMS - 0202508.The first proposed project is to understand the topology of complete manifolds whose Ricci curvature is bounded from below by a negative multiple of the bottom of the spectrum of the Laplace operator. In this case, the bottom of the spectrum is assumed to be positive. The second project of the Principal Investigator is to investigate the size of the space of harmonic functions with polynomial growth on complete manifolds with non-negative sectional curvature. In particular, the primary goal is to prove that the dimension of the space of harmonic functions of polynomial growth with degree at most d is bounded by the dimension of an analogous space in Euclidean space of the same dimension. The third project is to understand the geometry of stable minimal hypersurfaces of Euclidean space. More generally, he would like to study minimal hypersurfaces with have finite index. The PI also proposes to support a postdoc, Dr. Xiaofeng Sun, in his research projects. Dr. Sun proposes to study the regularity theory of harmonic maps into a metric space. These proposed projects have the unified theme of using analytical techniques in studying the underlining geometric spaces. In many instances, topological and geometrical conclusions can be drawn by detail analysis of solutions of some appropriate partial differential equations defined on the space.