Geometric Analysis on complete aspherical spaces

提案编号：DMS-0102552主要研究者研究完整的非球面空间，特别强调局部分裂结构的刚度和最小体积问题。曹博士打算继续他的工作，格罗莫夫的最小体积间隙猜想与他的合作者联合。利用Cheeger和Gromov发展的F-结构理论和热流，他想研究完备非球面流形的最小体积间隙猜想。研究者希望证明，如果紧致非正曲流形M同伦等价于广义图流形，则M也必是极小体积为零的广义图流形。 此外，曹计划继续研究紧致非球面流形的欧拉数的符号。本计画主要研究非球面空间的整体几何形状。非球面空间的例子包括轮胎漏气和表面有两个以上的孔，如椒盐卷饼。也有高维非球面空间的例子。我们的宇宙可以看作是一个三维的非球面空间。曹博士试图研究这些空间的直径、体积、光谱和其他几何数据。曹还对非正曲空间上的最短闭曲线的研究感兴趣。他已经表明，两个这样的表面可能的尖点等距当且仅当数据的长度最短的封闭曲线的两个表面是相同的。闭曲面M上所有最短闭曲线的长度数据称为空间M的标长谱。带边界空间上的标长谱的研究在现代工业和地质科学中有许多应用。

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.