Geometric Analysis on Semi-Hyperbolic Spaces with Variable Curvature

AbstractAward：DMS-0405385首席研究员：曹建国曹教授计划继续他的研究在几何分析的半双曲空间，特别强调非正曲流形的秩一。在所有的紧非正曲流形的秩一，principalinvestigator计划继续他的研究高维广义图流形。他想研究Gromov的最小体积间隙猜想，F-结构理论和非正曲流形的半刚性之间的关系。利用Cheeger和Gromov的F-结构理论，本文希望证明：如果流形M具有非正截面曲率度量，则M具有非零极小体积或M是极小体积为零的广义图流形，进而证明：如果M是紧致的非正曲率流形，且具有F-结构，则M确实是广义图流形。与Cheeger和Rong一起，主要研究者最近发现，如果一个非正截面曲率的紧致流形M同伦等价于具有F-结构的紧致流形，则M必须具有一个具有非平凡局部toriFactor的局部度量分裂结构。主要研究者希望继续他的联合研究项目与Cheeger和荣在这个方向。在与克罗克博士的合作中，首席研究员还计划继续他对测地线流的刚性和非正截面曲率流形的标记长度谱的研究。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流形的秩一和马丁边界。他对任何紧致Kaehler a-球面流形的Euler数的符号的研究也将继续进行。本项目主要研究非正曲空间的整体几何形状。非正弯曲空间的例子包括爆胎和有两个以上孔的表面，如椒盐卷饼。 也有高维非正曲空间的例子。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很感兴趣。他已经证明了两个可能有尖点的曲面是等距的当且仅当两个曲面上所有最短闭曲线的长度数据相同。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。所提出的问题涉及到黎曼几何和Kaehler几何的各个方面。这些方法与数学的其他领域有着密切的联系，包括拓扑学、偏微分方程、多复变函数和动力系统。这些问题的解决方案将推动所有这些错综复杂的相关领域的发展，并开辟一个广阔的未开发领域。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学科有着密切的联系。

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.