Global Analysis on Complete Kahler Manifolds

摘要奖：DMS-9970284主要研究人员：Ni主要研究完备Kahler流形上的几何和分析之间的相互作用。重点是完备Kahler型流形在各种曲率条件下的线性和非线性分析。主要目的是了解完备Kahler流形上在各种曲率假设下的全纯函数空间(以及某些全纯丛的截面)，并将这些空间的性质应用于研究具有各种曲率条件的完备Kahler流形的几何和拓扑，特别是具有正对分曲率的完备Kahler流形的均匀化。PI的研究课题属于流形上的微分几何和整体分析领域。微分几何是研究空间(微分几何语言中的流形)的几何和定义在该空间上的分析概念之间的关系的学科。全局分析是通过拼接局部信息来研究空间的整体几何和拓扑性质的学科。由于空间通常是一个弯曲的空间，引入了术语“曲率”来测量与欧几里得空间的偏差，欧几里德空间是平的。这些数学领域在其他科学中的应用包括复杂分子结构、气液边界和宇宙的大尺度结构。Kahler流形是一个复杂的空间，它有一个额外的结构叫做Kahler结构。Kahler流形的几何性质和解析性质对于低维拓扑、代数几何和超弦理论等数学和物理领域具有重要意义。

AbstractAward: DMS-9970284Principal Investigator: Lei NiThe principal investigator propose to study the interplay betweengeometry and analysis on complete Kahler manifolds. The focuswill be the linear and nonlinear analysis on complete Kahlermanifolds under various curvature conditions. The main goal is tounderstand the spaces of holomorphic functions (and sections ofcertain holomorphic bundles) on complete Kahler manifolds undervarious curvature assumptions and apply the properties of thesevarious spaces to the study of the geometry and topology ofcomplete Kahler manifolds with various curvature conditions, inparticular, the uniformization of complete Kahler manifolds withpositive bisectional curvature.The subject of the PI's investigation belongs to fields calleddifferential geometry and global analysis onmanifolds. Differential geometry is the study of the relationshipbetween the geometry of a space (a manifold in the language ofdifferential geometry) and analytic concepts defined on thespace. Global analysis is the study of the overall geometric andtopological properties of a space by piecing together localinformation. Since the space is usually a curved space, the term``curvature'' is introduced to measure the deviation from theEuclidean space, which is flat. Applications of these areas ofmathematics in other sciences include the structure ofcomplicated molecules, liquid-gas boundaries, and the large scalestructure of the universe. A Kahler manifold is a complex spacewhich has an extra structure called Kahler structure. Thegeometric and analytic properties of Kahler manifolds areimportant to various fields of mathematics and physics such aslow dimensional topology, algebraic geometry and superstringtheory.