Global Riemannian Geometry and Analysis of curved spaces

曹教授计划继续他在全球黎曼几何和曲线空间分析方面的研究，重点是非正曲率的流形。他将继续使用(可能是奇异的)微分方程，以解决黎曼几何中的各种问题。此外，PI继续通过黎曼几何和次黎曼几何的方法研究极小正调和函数和CR-爱因斯坦方程。求解微分方程的许多重要进展依赖于对这些问题的几何理解。国际和平研究所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将使用分析中的技巧来研究黎曼几何和柯西-黎曼几何中的各种问题，与Cheeger-Rong一起，PI打算使用各种分叶热流来研究非正曲率流形上的Cheeger-Gromov折叠理论。他希望了解非正曲线流形上所谓的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。特别地，他想部分地回答Yau关于极小正调和函数的一个问题(即Ballmann一阶流形的Martin边界问题)。第三，PI计划通过各种几何方法研究次黎曼流形上的CR-爱因斯坦方程和CR-Calabi问题。此外，PI还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。他将研究距离函数的广义Morse理论和3-流形的折叠之间的关系，正如Perelman最近的工作所概述的那样。

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.