Geometric Analysis on complete aspherical spaces

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Proposal Number: DMS-0102552The principal investigator studies complete aspherical spaces with particular emphasis on the rigidity of local splitting structures and the minimal volume problem. Dr. Cao intends to continue his work on Gromov's minimal volume gap conjecture jointly with his coauthors. Using the F-structure theory developed by Cheeger and Gromov and the heat flow, he would like to study the minimal volume gap conjecture for complete aspherical manifolds. The investigator hopes to show that if a compact nonpositively curved manifold $M$ is homotopy equivalent to a generalized graph-manifold, then $M$ must be a generalized graph-manifold with vanishing minimal volume as well. In addition, Cao plans to continue his study of the sign of the Euler number of compact aspherical manifolds. This project focuses on the study of global geometric shape of aspherical spaces. The examples of aspherical spaces include flat tires and surfaces with more than two holes, such as pretzels. There are also examples of higher dimensional aspherical spaces. Our universe can be viewed a 3-dimensional aspherical space. 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.