Parabolic Equations and the Geometry of Complete Kaehler Manifolds

AbstractAward：DMS-0504792首席研究员：倪磊首席研究员提出研究完备Kaehler流形上几何与分析之间的相互作用。重点将是完备流形上的线性和非线性抛物型方程的各种曲率假设。这些方程包括线性热方程和Laplacian算子、调和映射热方程、平均曲率流、Hermitian-Einstein流、Ricci/Kaehler-Ricciflow等。 但它们有许多共同的性质，如几何对称性和恒等式，单调性公式，熵似的考虑，微分Harnack（也叫Li-Yau-汉密尔顿）不等式等。 它们也以各种方式相互连接。 微分几何是一门研究空间（数学概念中的流形）的几何学与函数和微分方程的解析性质之间关系的学科。几何分析是将空间的局部信息拼接起来，研究空间的整体几何和拓扑性质。由于空间通常是弯曲的，“曲率”被引入来测量与欧几里得空间的偏差，并且即使处理线性问题，该技术也经常是“非线性”的。这一数学领域的研究与物理学中的广义相对论、弦理论有着密切的联系。它可以应用于复杂分子结构、气液边界、甚至大型不对称网络的研究。本计画借由线性与非线性抛物型方程来研究Kaehler流形，将增进对几何分析、线性与非线性偏微分方程、代数几何与物理的了解。

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.