Differential Geometry by way of Partial Differential Equations

摘要DMS - 0202508。第一个提议的项目是理解完全流形的拓扑结构，其里奇曲率从下到下由拉普拉斯算子谱底的负倍数限定。在这种情况下，假定频谱的底部是正的。本课题的第二个课题是研究非负截面曲率的完全流形上多项式增长的调和函数的空间大小。特别地，主要目的是证明次数最多为d的多项式生长的调和函数空间的维数被相同维数的欧几里得空间中的类似空间的维数有界。第三个课题是了解欧氏空间稳定极小超曲面的几何。更一般地说，他想研究具有有限指数的极小超曲面。PI还建议支持博士后孙晓峰博士的研究项目。孙博士建议研究度量空间调和映射的正则性理论。这些提议的项目有一个统一的主题，即使用分析技术来研究突出的几何空间。在许多情况下，通过详细分析空间上定义的一些适当的偏微分方程的解，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.