Global Riemannian Geometry and Analysis of curved spaces

全局黎曼几何与弯曲空间分析

• 批准号: 0706513
• 负责人: Jianguo Cao
• 金额: $ 10.77万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706513&HistoricalAwards=false
• 关键词: Global; Riemannian; Geometry; Analysis; curved

Professor Cao plans to continue his research on global Riemannian geometry and analysis of curved spaces, with emphasis on manifolds of non-positive curvature. He will continue the use of (possibly singular) differential equations in order to solve various problems in Riemannian geometry. In addition, the PI continues to study minimal positive harmonic functions and CR-Einstein equations via methods from Riemannian geometry and sub-Riemannian geometry. Many of the important advances in solving differential equations depend on the geometric understanding of these problems. The PI will continue to work in this direction. Solving problems in global Riemannian geometry sometimes depends on new tools from analysis. The PI will use techniques from analysis to investigate various problems in Riemannian geometry and Cauchy-Riemann geometry.Together with Cheeger-Rong, the PI intends to use various foliated heat flows to study the Cheeger-Gromov collapsing theory on manifolds with non-positive curvature. He hopes to understand the homotopy invariance of so-called F-structures on non-positively curved manifolds. Secondly, the PI and his collaborators will continue the study of minimal positive harmonic functions on open manifolds. In particular, he wants to provide a partial answer to a problem of Yau on minimal positive harmonic functions (i.e., Martin boundary problem for manifolds of Ballmann rank one). Thirdly, the PI plans to study the CR-Einstein equations and the CR-Calabi problem on sub-Riemannian manifolds via various geometric methods. In addition, the PI plans to continue the study of the critical point theory for distance function on Alexandrov spaces with curvature bounded below. He will investigate the relations between generalized Morse theory for distance functions and the collapsing of 3-manifolds, as outlined in Perelman's recent work.

曹教授计划继续研究整体黎曼几何和弯曲空间的分析，重点是非正曲率流形。他将继续使用（可能奇异）微分方程，以解决各种问题的黎曼几何。此外，PI继续通过黎曼几何和亚黎曼几何的方法研究极小正调和函数和CR-Einstein方程。解微分方程的许多重要进展都依赖于对这些问题的几何理解。PI将继续朝着这个方向努力。解决整体黎曼几何中的问题有时依赖于来自分析的新工具。PI将使用分析技术来研究黎曼几何和Cauchy-Riemann几何中的各种问题。与Cheeger-Rong一起，PI打算使用各种叶状热流来研究非正曲率流形上的Cheeger-Gromov塌缩理论。他希望了解非正弯曲流形上所谓的F结构的同伦不变性。其次，PI和他的合作者将继续研究开流形上的极小正调和函数。特别是，他想提供一个部分的答案，以最小的正调和函数（即，Martin边界问题的Ballmann秩一）。第三，PI计划通过各种几何方法研究次黎曼流形上的CR-Einstein方程和CR-Calabi问题。此外，PI计划继续研究曲率有界的Alexandrov空间上距离函数的临界点理论。他将调查之间的关系广义莫尔斯理论的距离函数和崩溃的3流形，概述了佩雷尔曼最近的工作。