Geometric Analysis on Semi-Hyperbolic Spaces with Variable Curvature

项目负责人：曹建国曹教授计划在半双曲空间的几何分析方面继续进行研究，重点是1阶非正弯曲流形。在所有1阶紧非正弯曲流形中，主要研究者计划继续他对高维广义图流形的研究。他想研究Gromov最小体积间隙猜想、f结构理论和非正弯曲流形的半刚性之间的关系。利用Cheeger和Gromov发展的f结构理论，希望证明如果amamfold M允许非正截面曲率度规，则M具有不消失的最小体积或M是具有零最小体积的广义图流形。进一步，作者试图证明，如果M是一个紧非正弯曲流形，且允许f结构，则M确实是一个广义图流形。最近与Cheeger和Rong共同发现，如果非正截面曲率的紧流形M同伦等价于具有f结构的紧流形，则M必须具有具有非平凡局部因子的局部分裂结构。首席研究员希望在这个方向上继续他与Cheeger和Rong的联合研究项目。在与克罗克博士的合作中，首席研究员还计划继续研究非正截面曲率流形的测地线流的刚性和标记长度谱。他想证明，如果一对非正截面曲率的紧致广义图流形具有相同的标记长度-谱，那么它们一定是等距的。此外，PI想继续他的研究正调和函数在流形上的秩1和马丁边界。他对任何紧化Kaehler a-球流形的欧拉数符号的研究也将继续。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。也有一些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，当且仅当两个曲面上所有最短闭合曲线的长度数据相同时，两个可能有顶点的曲面是等距的。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。对有边界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。提出的问题涉及黎曼几何和凯勒几何的各个方面。在提议的研究中开发的技术将与其他数学领域有密切的联系，包括拓扑，偏微分方程，几个复杂变量和动力系统。提出的问题的解决方案将推进所有这些错综复杂的相关领域，并开辟一个广阔的未开发领域。所提出的标记长度谱和测地线流的研究与包括地球科学在内的其他科学分支密切相关。

AbstractAward: DMS-0405385Principal Investigator: Jianguo CaoProfessor Cao plans to continue his research on the geometricanalysis of semi-hyperbolic spaces with a particular emphasis onnon-positively curved manifolds of rank one. Among all compactnon-positively curved manifolds of rank one, the principalinvestigator plans to continue his study of the higherdimensional generalized graph-manifolds. He would like toinvestigate relations among Gromov's minimal volume gapconjecture, F-structure theory and semi-rigidity fornon-positively curved manifolds. Using the F-structure theorydeveloped by Cheeger and Gromov, the hopes to show that if amanifold M admits a metric of non-positive sectional curvature,then either M has non-vanishing minimal volume or M is ageneralized 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 anF-structure, then M indeed is a generalizedgraph-manifold. Together with Cheeger and Rong, the principalinvestigator recently discovered that, if a compact manifold M ofnon-positive sectional curvature is homotopy equivalent to acompact manifold with an F-structure, then M must have a localmetric splitting structure with nontrivial local torifactors. The principal investigator would like to continue hisjoint research project with Cheeger and Rong in thisdirection. In cooperation with Dr. Croke, the principalinvestigator also plans to continue his study of rigidity of thegeodesic flow and marked length-spectrum for manifolds ofnon-positive sectional curvature. He would like to show that, ifa pair of compact generalized graph-manifolds of non-positivesectional curvature have the same marked length-spectrum, thenthey must be isometric. In addition, the PI would like tocontinue his study of positive harmonic functions on manifolds ofrank one and Martin boundary. His research on the sign of theEuler number of any compact Kaehler a-spherical manifold willalso be continued.This project focuses on the study of global geometric shape ofnon-positively curved spaces. The examples of non-positivelycurved spaces include flat tires and surfaces with more than twoholes, such as pretzels. There are also examples of higherdimensional non-positively curved spaces. Our universe can beviewed a 3-dimensional space of zero curvature. Dr. Cao is tryingto investigate diameter, volume, spectrum and other geometricdata of those spaces. Cao has also been interested in the studyof the shortest closed curves on non-positively curved spaces. Hehas already shown that two such surfaces with possible cusps areisometric if and only if the data of lengths of all shortestclosed curves on the two surfaces are identical. The data oflengths of all shortest closed curves on a closed surface M iscalled the marked length spectrum of the space M. The study ofmarked length spectrum on spaces with boundaries has a number ofapplications in modern industry and geological sciences. Theproposed problems involve various aspects of Riemannian geometryand Kaehler geometry. The techniques developed in the proposedresearch will have close connections with other fields inmathematics, including topology, partial differential equations,several complex variables and dynamical systems. The solutions tothe proposed problems will advance all these intricately relatedfields and open up a vast unexplored area. The proposed study ofthe marked length-spectrum and the geodesic flow has closeconnections with other branches of sciences includinggeosciences.