Parabolic Equations and the Geometry of Complete Kaehler Manifolds

摘要：项目负责人：倪磊，主要研究完全kaehler流形几何与分析之间的相互作用。重点将是具有各种曲率假设的完全流形上的线性和非线性抛物方程。这些方程包括线性热方程和拉普拉斯算子、调和映射热方程、平均温度流、埃尔米特-爱因斯坦流、里奇/凯勒-里奇流等。这些方程有不同的物理和几何根源。但它们有许多共同的特征，如几何对称性和恒等式，单调性公式，熵样考虑，微分哈纳克（也称为李尧-汉密尔顿）不等式。它们也以各种方式相互连接。通过一起研究它们，更多的光被照射到它们身上。微分几何是研究空间的几何关系，数学概念中的流形，以及底层空间上的函数和微分方程的解析性质。几何分析是通过拼凑局部信息来研究空间的整体几何和拓扑性质。由于空间通常是弯曲的，因此引入“曲率”来测量与欧几里得空间的偏差，即使处理线性问题，技术也往往是“非线性的”。这一数学领域的研究与物理学中的广义相对论和弦理论有着密切的联系。应用于复杂分子结构、液气边界、甚至大规模网络的研究。本课题对Kaehler流形的线性和非线性抛物方程的研究将加强对几何分析、线性和非线性偏微分方程、代数几何和数学物理的理解。

AbstractAward: DMS-0504792Principal Investigator: Lei NiThe principle investigator proposes to study the interplaybetween the geometry and the analysis on complete Kaehlermanifolds. The focus will be the linear and nonlinear parabolicequations on complete manifolds with various curvatureassumptions. These equations includes the linear heat equationand Laplacian operator, harmonic mapping heat equation, meancurvature flow, Hermitian-Einstein flow, Ricci/Kaehler-Ricciflow, etc. These equations have various physical and geometricorigins. But they share many common features such as geometricsymmetries and identities, monotonicity formulae, entropy likeconsiderations, differential Harnack (also calledLi-Yau-Hamilton) inequalities. They also connect to each otherin various ways. By studying them together, more light is shedon all of them.Differential Geometry is the study of the relationship betweenthe geometry of a space, a manifold in the mathematical notion,and the analytic properties of the functions and the differentialequations, on the underlying space. Geometric analysis is thestudy of the overall geometric and topological properties of aspace by piecing together the local information. Since the spacesare usually curved ones, the ``curvature" was introduced tomeasure the deviation from the Euclidean space and the techniquesare often `nonlinear' even dealing with a linear problem. Thestudy of this area of mathematics has close connection with thegeneral relativity and string theory in physics. The applicationscan be found in the study of thestructure of complicatedmolecules, liquid-gas boundaries, and even the large scalenetworks. This project on studying the Kaehler manifolds vialinear and nonlinear parabolic equations will enhance theunderstanding of geometric analysis, linear and nonlinear partialdifferential equations, algebraic geometry and mathematicalphysics.