Analysis on Complete Manifolds

AbstractAward：DMS-9971418首席研究员：Peter Li在这个项目中，首席研究员建议研究与其几何结构相关的完备流形的各种分析方面。 特别是，他提出要理解空间的多项式增长调和函数的一般类的流形，其中包括流形与非负Ricci曲率，也极小子流形的欧氏空间，其中有欧氏体积增长。 他还提出了理解调和映射的图像集的结构及其对底层几何结构、拓扑结构和复杂结构的含义。 最后，他提出研究具有足够小的平均值常数的整体平均值不等式的有效性的影响。几何分析结构的研究是数学发展中的一个基本而重要的部分。 它也直接影响到科学和工程的许多分支，如广义相对论、机器人学、材料科学和计算机图形学。

AbstractAward: DMS-9971418Principal Investigator: Peter LiIn this project, the principal investigator proposes to studyvarious analytical aspects of a complete manifold in relation toits geometrical structure. In particular, he proposes tounderstand the space of polynomial growth harmonic functions on ageneral class of manifolds that includes manifolds withnon-negative Ricci curvature and also minimal submanifolds ofEuclidean space which has Euclidean volume growth. He alsoproposes to understand the structure of the image set of harmonicmaps and it implications to the underlining geometric,topological, and complex structures. Finally, he proposes toinvestigate the impact of the validity of a global mean valueinequality with sufficiently small mean value constant.The study of geometric-analytical structure is a fundamental andimportant part in the development of mathematics. It also hasdirect implications to many branches of sciences and engineering,such as, general relativity, robotics, material science, andcomputer graphics.