Linear and Nonlinear Analysis on Complete Kahler Manifolds

本文的主要研究者提出研究完备Kaehler流形上几何与分析之间的相互作用。重点将是对这类流形的非线性和非线性分析。其主要工具是求解线性方程，如Poisson方程和Poincare-Lelong方程，以及非线性方程，如Kaehle-Ricci流。目标是理解全纯函数空间（复数次调和函数），几何和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备Kaehler流形的一致化。流形上的整体分析是通过将局部信息拼接在一起来研究流形的整体性质。Kaehler流形是弦论宇宙模型的基本块。本文的研究与广义相对论和弦理论有着密切的联系。该方案所研究的非线性微分方程在复杂分子结构、液-气边界、甚至大规模网络的研究中具有应用价值。

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.