Global Riemannian Geometry and Analysis of curved spaces

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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.