Global Analysis on Complete Kahler Manifolds

AbstractAward：DMS-9970284主要研究者：倪磊主要研究者提出研究完全Kahler流形上的几何测量和分析之间的相互作用。重点讨论了完备Kahler流形在各种曲率条件下的线性和非线性分析。主要目的是理解全纯函数空间（以及某些全纯丛的截面），并将各种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流形的一致化。PI研究的主题属于称为微分几何的领域，流形上的整体分析微分几何是研究空间几何（微分几何语言中的流形）和空间上定义的分析概念之间关系的学科。全局分析是通过将局部信息拼接在一起来研究空间的整体几何和拓扑性质。由于空间通常是一个弯曲的空间，术语“曲率”被引入来衡量偏离欧几里得空间，这是平坦的。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。一个Kahler流形是一个复空间，它有一个额外的结构，称为Kahler结构。Kahler流形的几何和解析性质在低维拓扑学、代数几何和超弦理论等数学和物理学的各个领域都有重要意义。

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.