Differential Geometry by way of Partial Differential Equations

通过偏微分方程的微分几何

• 批准号: 0202508
• 负责人: Peter Li
• 金额: $ 22.5万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202508&HistoricalAwards=false
• 关键词: Differential; Geometry; way; Partial; Equations

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.

摘要DMS -0202508.第一个提出的项目是理解完全流形的拓扑结构，其Ricci曲率由拉普拉斯算子的谱底的负倍数从下有界。 在这种情况下，假设频谱的底部为正。 主要研究者的第二个项目是研究具有非负截面曲率的完备流形上多项式增长的调和函数空间的大小。 特别是，主要的目标是证明，与度最多d的多项式增长的调和函数的空间的尺寸是由一个类似的空间在相同尺寸的欧几里得空间的尺寸有界。 第三个项目是了解欧几里得空间的稳定极小超曲面的几何。更一般地说，他想研究极小超曲面有有限的指数。 PI还建议支持博士后孙晓峰博士的研究项目。 孙博士提出研究度量空间中调和映射的正则性理论。这些拟议的项目有一个统一的主题，即在研究底层几何空间时使用分析技术。 在许多情况下，拓扑和几何的结论可以通过详细分析的解决方案，一些适当的偏微分方程定义在空间。