Abstract Proposal Number: DMS-9803230 Principal Investigator: Jianguo Cao The principal investigator studies manifolds of non-positive sectional curvature with particular emphasis on isoperimetric inequalities 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, he would like to study the minimal volume gap conjecture for complete aspherical manifolds. Cao also plans to continue his study on isoperimetric inequalities on simply-connected riemannian manifolds of non-positive sectional curvature. As an intermediate step towards understanding this isoperimetric problem, he hopes to study the total Gauss-Kronecker curvature for smooth compact convex hypersurfaces in Cartan-Hadamard manifolds. In addition, Cao plans to continue his study of the sign of the Euler number of compact aspherical manifolds, especially in the complex case. 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 geologica l sciences. In addition, the research of spaces of dimension 10 and 26 have played a crucial role in the theoretical physics of the unification of four fundamental forces in our universe.
