Global Analysis on Complete Kahler Manifolds

项目编号：dms -9970284项目负责人：倪磊，主要研究完全Kahler流形的几何与分析之间的相互作用。重点将是在不同曲率条件下完全kahler流形的线性和非线性分析。主要目的是了解不同曲率条件下完全Kahler流形上全纯函数的空间（以及某些全纯束的截面），并将这些空间的性质应用于研究具有不同曲率条件的完全Kahler流形的几何和拓扑，特别是具有正对分曲率的完全Kahler流形的均匀化。PI调查的主题属于流形的微分几何和全局分析领域。微分几何是研究空间的几何（用微分几何的语言称为流形）与空间上定义的解析概念之间关系的学科。全局分析是通过拼凑局部信息来研究空间的整体几何和拓扑特性。由于空间通常是弯曲空间，因此引入“曲率”一词来测量与欧几里得空间的偏差，欧几里得空间是平坦的。这些数学领域在其他科学中的应用包括复杂分子的结构、液气边界和宇宙的大尺度结构。卡勒流形是一个复杂的空间，它有一个额外的结构叫做卡勒结构。Kahler流形的几何和解析性质对数学和物理的各个领域都很重要，如慢维拓扑、代数几何和超弦理论。

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.