Linear and Nonlinear Analysis on Complete Kahler Manifolds

摘要DMS-0203023.PI：Lei Ni主要研究者建议研究几何与完备Kaehler流形分析之间的相互作用。重点将对这类流形进行线性和非线性分析。主要工具是求解线性方程，如泊松方程、庞加莱-拉朗方程，以及非线性方程，如Kaehle-Ricci流。其目的是了解全纯函数(多重次调和函数)的空间，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。流形是发生任何物理事件的空间。流形上的全局分析通过拼接局部信息来研究流形的整体性质。根据弦理论，Kaehler流形是宇宙模型中的基本块。提出的研究与广义相对论和弦理论有着密切的联系。所研究的非线性微分方程组在研究复杂分子结构、气液界面乃至大规模网络中具有重要的应用价值。

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.