Analysis on Complete Manifolds

摘要奖：DMS-9971418首席研究员：Peter Lie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整流形的各种分析方面。特别地，他建议理解一般流形上的多项式增长调和函数空间，这些流形包括具有非负Ricci曲率的流形和具有欧氏体积增长的欧氏空间的极小子流形。他还建议理解调和映射图像集的结构，以及它对下面的几何、拓扑和复杂结构的影响。最后，他建议研究一个具有足够小的平均值常数的全局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和重要的部分。它还对许多科学和工程分支有直接影响，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.