Linear and Nonlinear Analysis on Complete Kahler Manifolds

完全卡勒流形的线性和非线性分析

• 批准号: 0328624
• 负责人: Lei Ni
• 金额: $ 5.87万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-09-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0328624&HistoricalAwards=false
• 关键词: Linear; Nonlinear; Analysis; Complete; Kahler

ABSTRACT DMS - 0203023.PI: Lei NiThe principle investigator proposes to study the interplay between thegeometry and the analysis on complete Kaehler manifolds. The focus will belinear and nonlinear analysis on such manifolds. The main tool is solvingthe linear equation, such as the Poisson equation and Poincare-Lelongequation, and the nonlinear equations such as the Kaehle-Ricci flow. Thegoal is to understand the space of holomorphic functions (plurisubharmonicfunctions), the interplay between the geometry and the function theory andapplying the results to the uniformization of complete Kaehler manifoldswith nonnegative curvature.The manifold is the space where every physical event happens. Theglobal analysis on manifolds studies the overall properties of themanifolds by piecing together the local information. Kaehler manifolds arethe basic block in the universe model according to the string theory. The proposed studyhas close connection with the theory of general relativity and stringtheory. The nonlinear differential equations studied in the proposal haveapplications in the study of the structure of complicated molecules,liquid-gas boundary, and even the large scale networks.

摘要DMS - 0203023。倪磊主要研究完全Kaehler流形的几何与分析之间的相互作用。重点将放在这种流形的线性和非线性分析上。主要的工具是求解线性方程，如泊松方程和庞加莱方程，以及非线性方程，如Kaehle-Ricci流。目的是了解全纯函数（多次谐波函数）的空间，几何与函数理论之间的相互作用，并将结果应用于具有非负曲率的完全Kaehler流形的均匀化。流形是每个物理事件发生的空间。流形的全局分析是将流形的局部信息拼凑在一起，研究流形的整体特性。根据弦理论，凯勒流形是宇宙模型中的基本块。所提出的研究与广义相对论和弦理论有着密切的联系。本文所研究的非线性微分方程在复杂分子结构、液气边界甚至大型网络的研究中具有广泛的应用价值。