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了解完整流形的拓扑，其 Ricci 曲率从下方以拉普拉斯算子谱底部的负倍数为界。 在这种情况下，假设频谱的底部为正。 首席研究员的第二个项目是研究具有非负截面曲率的完全流形上多项式增长的调和函数空间的大小。 特别是，主要目标是证明次数最多为 d 的多项式增长的调和函数空间的维数受相同维数的欧几里得空间中类似空间的维数的限制。 第三个项目是了解欧几里德空间稳定最小超曲面的几何形状。更一般地说，他想研究具有有限索引的最小超曲面。 PI 还提议支持博士后孙晓峰博士的研究项目。 孙博士建议研究调和映射到度量空间的正则理论。这些拟议项目的统一主题是使用分析技术来研究底层几何空间。 在许多情况下，可以通过对空间上定义的一些适当的偏微分方程的解进行详细分析来得出拓扑和几何结论。