Mathematical Sciences: Harmonic Maps and Harmonic Functions

9626310 Li建议的研究涉及调和函数，以及更一般的调和映射。三个相关的主题是：Hadamard流形之间的调和映射，可能奇异代数簇的调和映射，以及完备流形上的调和函数。特别地，由于Yau的猜想：对于任意非负整数k，k阶多项式增长的调和函数集在具有非负Ricci曲率的流形上一定是有限维的。调和函数作为复值解析函数的分量出现；因此，它们具有丰富的结构理论。在应用中，它们可以作为各种经典偏微分方程解的形式出现，并且可以用来描述薄层上的热流、张力和面积最小化的表面构型等。

9626310 Li The proposed research deals with harmonic functions, and more generally, harmonic maps. Three related topics are to be pursued: harmonic maps between Hadamard manifolds, harmonic maps from possibly singular algebraic varieties, and harmonic functions on complete manifolds. In particular, the investigator aims to work on the following conjecture due to Yau: the set of harmonic functions of k-th order polynomial growth, for any non-negative integer k, must be of finite dimension on a manifold with non-negative Ricci curvature. Harmonic functions appear as components of complex-valued analytic functions; as such they have a rich structure theory. In applications, they arise as solutions to various classical partial differential equations, and can be used to describe the heat flow on a lamina, tension and area-minimizing surface configurations, and so forth.