完备非紧黎曼流形的基本群

It is well known that the fundamental group of compact manifold is finitely generated. For complete non-compact Riemannian manifold, a natural question is: when its fundamental group is finitely generated? Milnor conjectured the fundamental group of any Riemannian manifolds with non-negative Ricci curvature is finitely generated. Recently, Kapovitch and Wilking used the theory of Ricci limit space and intrinsic Reifenberg theorem, succeeded in proving that for any n-dim manifolds with non-negative Ricci curvature, any finitely generated subgroup of the fundamental group can be generated by C(n) generators, where C(n) is a constant depending only on n. Based on the external theory of Ricci limit spaces, the argument of Kapovitch-Wilking is not intrinsic from the view point of Riemannian geometry. We expect to study the fundamental group of complete manifolds with non-negative Ricci curvature from the intrinsic view point, instead of apply the theory of Ricci limit spaces, try to use the analysis of the distance functions and harmonic functions on manifolds, to reveal the relationship between the algebraic structure of fundamental groups and the Ricci curvature of the manifolds more directly, and hope to give a complete proof of the quantitative version of intrinsic Reifenberg theorem.

众所周知紧流形的基本群是有限生成的，对于完备非紧的黎曼流形，一个自然的问题是：基本群何时是有限生成的？对于具有非负Ricci曲率的完备非紧流形，Milnor猜想其基本群是有限生成的。最近Kapovitch-Wilking证明了对于具有非负Ricci曲率的n维流形，其基本群的任意有限生成子群一定可以由C(n)个元素生成，C(n)是一个只依赖于n的常数。因为需要借助外在的Ricci极限空间理论和内蕴Reifenberg定理, Kapovitch-Wilking的证明从黎曼几何的角度来看并不是内蕴的。我们期望从内蕴的角度研究具有非负Ricci曲率的黎曼流形上的基本群，通过流形上的距离函数及其调和化后得到的调和函数这些分析工具，更直接的揭示基本群的代数结构和流形的Ricci曲率之间的关系，并且希望给出量化的内蕴Reifenberg定理的完整证明。

从20世纪下半叶以来，完备非紧流形上的几何拓扑以及调和函数论一直是几何分析中的.重要研究方向。对于具有非负Ricci曲率的完备非紧流形，Milnor猜测其基本群是有限生成的。该猜想已经在3维时通过对此类流形的拓扑分类得到解决。对高维的情形，基于Kapovitch-Wilking之前的工作，本项目以具有非负Ricci曲率的完备非紧流形上的基本群为.研究对象，建立此类流形上基本群生成元的个数以及局部调和函数的分析之间的直接联系。.我们证明了有限生成子群生成元个数的一致显式估计，并且得到在特定情形下调和函数的频率以及最佳梯度估计。本项目揭示了非负Ricci曲率的完备非紧流形上调和.函数和该流形的几何拓扑性质的关系，从而为此类流形上的几何拓扑以及调和函数的研究提供技术工具以及深化认知。