Geometric Analysis on Manifolds of Non-positive Curvature

项目编号：DMS-9803230项目负责人：曹建国主要研究非正截面曲率流形，重点研究等周不等式和最小体积问题。曹博士打算与他的合作者一起继续他对Gromov最小体积间隙猜想的研究。利用Cheeger和Gromov的f结构理论，研究完全非球面流形的最小体积间隙猜想。曹还计划继续研究非正截面曲率单连通黎曼流形上的等周不等式。作为理解这个等周问题的中间步骤，他希望研究Cartan-Hadamard流形中光滑紧致凸超曲面的总高斯-克罗内克曲率。此外，曹计划继续他的紧非球流形的欧拉数符号的研究，特别是在复杂情况下。本项目主要研究非正弯曲空间的整体几何形状。非正弯曲空间的例子包括扁平的轮胎和有两个以上洞的表面，比如椒盐脆饼。也有高维非正弯曲空间的例子。我们的宇宙可以看作是一个零曲率的三维空间。曹博士正试图研究这些空间的直径、体积、光谱和其他几何数据。曹还对非正弯曲空间上的最短闭曲线的研究感兴趣。他已经证明，两个有可能顶点的曲面是等距的，当且仅当两个曲面上所有最短闭合曲线的长度数据相同。在一个封闭曲面M上所有最短的封闭曲线的长度数据称为空间M的标记长度谱。对具有边界的空间上的标记长度谱的研究在现代工业和地质科学中有许多应用。此外，10维和26维空间的研究在宇宙中四种基本力统一的理论物理中起着至关重要的作用。

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.