非正曲率流形上的有界调和函数

非正曲率的完备单连通的黎曼流形上有界调和函数的存在性问题是整体微分几何和几何分析研究中一个倍受人们关注的问题（参见S.T. Yau于1993年提出的第二个120问题集中的第47问题[1](文献在项目的立项依据中所列)），这既是探究这类流形（包括复流形）几何性质的一种重要的手段，也与复几何中的一些基本问题密切相关。本项目拟在申请人就这一问题的研究所取得一定进展（参见[9](在立项依据中所列)）的基础上，作进一步的探索和研究，以期在该问题及有关问题的研究上能够取得更大的突破，并且也尝试把这种探索和研究推广到某种Alexandrov空间。

The existence of nontrivial bounded harmonic functions on complete simply-connected Riemannian manifolds of nonpositive sectional curvature is an attractive problem in global differential geometry and geometric analysis (refer to the No.47 problem in the second collection of 120 problems by S.T. Yau [S.T. Yau, Open problems in geometry, in Differential Geometry: Partial differential equations on manifolds (Los Angeles, CA 1990), 1-28, ed. by R. Greens and S.T. Yau, Proc. Symps. Pure Math. Vol 54 Part I, AMS, Providence, 1993.]). The study is a right way to understand the deep geometric properties of Riemannian manifolds (including complex manifolds) with nonpositive sectional curvature and also relates to some fnndamental problems in complex geometry. Based on the recent work done by the applicant (see [Q. Ding, Bounded harmonic functions on Riemannian manifolds of non-positive curvature, Math. Ann. Published online 02 August 2011 (DOI10.1007/ s00208-011-0705-9).]), we aim to continue the exploration on this problem. It is expected that a further progress will be achieved during the period of performing the proposal. We also attempt to generalize such a study to Alexandrov spaces.