Mathematical Sciences: Harmonic Maps and Harmonic Functions

小行星9626310 拟议的研究涉及调和函数，更一般地说，调和映射。三个相关的主题是追求：调和映射之间的阿达玛流形，调和映射可能奇异代数品种，调和函数的完整流形。特别是，调查的目的是工作在以下猜想由于丘：一组调和函数的k阶多项式增长，任何非负整数k，必须是有限维的流形上的非负里奇曲率。 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，它们作为各种经典偏微分方程的解出现，并且可以用于描述层流、张力和面积最小化表面配置等上的热流。

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.