Principal Investigator: Jianguo Cao
Professor Cao plans to continue his research on the geometric analysis of semi-hyperbolic spaces with a particular emphasis on non-positively curved manifolds of rank one. Among all compact non-positively curved manifolds of rank one, the principal investigator plans to continue his study of the higher dimensional generalized graph-manifolds. He would like to investigate relations among Gromov's minimal volume gap conjecture, F-structure theory and semi-rigidity for non-positively curved manifolds. Using the F-structure theory developed by Cheeger and Gromov, the hopes to show that if a manifold M admits a metric of non-positive sectional curvature, then either M has non-vanishing minimal volume or M is a generalized graph-manifold with zero minimal volume. Furthermore, the principal investigator intends to verify that, if M is a compact non-positively curved manifold, which admits an F-structure, then M indeed is a generalized graph-manifold. Together with Cheeger and Rong, the principal investigator recently discovered that, if a compact manifold M of non-positive sectional curvature is homotopy equivalent to a compact manifold with an F-structure, then M must have a local metric splitting structure with nontrivial local tori factors. The principal investigator would like to continue his joint research project with Cheeger and Rong in this direction. In cooperation with Dr. Croke, the principal investigator also plans to continue his study of rigidity of the geodesic flow and marked length-spectrum for manifolds of non-positive sectional curvature. He would like to show that, if a pair of compact generalized graph-manifolds of non-positive sectional curvature have the same marked length-spectrum, then they must be isometric. In addition, the PI would like to continue his study of positive harmonic functions on manifolds of rank one and Martin boundary. His research on the sign of the Euler number of any compact Kaehler a-spherical manifold will also be continued.
This project focuses on the study of global geometric shape of non-positively curved spaces. The examples of non-positively curved spaces include flat tires and surfaces with more than two holes, such as pretzels. There are also examples of higher dimensional non-positively curved spaces. Our universe can be viewed a 3-dimensional space of zero curvature. Dr. Cao is trying to investigate diameter, volume, spectrum and other geometric data of those spaces. Cao has also been interested in the study of the shortest closed curves on non-positively curved spaces. He has already shown that two such surfaces with possible cusps are isometric if and only if the data of lengths of all shortest closed curves on the two surfaces are identical. The data of lengths of all shortest closed curves on a closed surface M is called the marked length spectrum of the space M. The study of marked length spectrum on spaces with boundaries has a number of applications in modern industry and geological sciences. The proposed problems involve various aspects of Riemannian geometry and Kaehler geometry. The techniques developed in the proposed research will have close connections with other fields in mathematics, including topology, partial differential equations, several complex variables and dynamical systems. The solutions to the proposed problems will advance all these intricately related fields and open up a vast unexplored area. The proposed study of the marked length-spectrum and the geodesic flow has close connections with other branches of sciences including geosciences.
