双曲流形的体积与形变

The hyperbolic manifold is an important research subject in complex analysis. Recently, there are a series of results on its classification, local structures and topological properties. Based on the known work, we will mainly discuss the volumes and deformations of hyperbolic manifolds, the detailed research contents are as follows: (1) discussing the properties of hyperbolic isometries and giving a lower bound on the radiuses of hyperbolic balls embedded in complex hyperbolic manifolds; (2) studying the discreteness of convergence groups and completing the discussion on the convergence of negatively curved groups acting on higher dimensional hyperbolic manifolds and Hadamard manifolds. The research of this project has great theory significance in strengthening the relationship between real and complex hyperbolic manifolds and Kleinian groups.

双曲流形是复分析领域中的一个重要研究课题。近年来，关于它的分类、局部结构以及拓扑性质已有一系列的研究成果。基于前人的工作，我们将主要围绕双曲流形的体积及其形变展开研究，具体内容如下：(1) 讨论双曲等距映射的性质，给出复双曲流形的内嵌球一致半径的下界；(2) 研究收敛群的离散性，进一步完善高维双曲流形、Hadamard流形上的负曲率等距群收敛性的讨论。本项目的研究加强了实、复双曲流形与Klein群的联系，具有重要的理论意义。

本项目主要研究双曲流形的体积与形变问题。目前，已基本完成该项目制定的研究计划, 在双曲流形、Hadamard流形的体积与形变以及Mobius群的离散性等方面取得了一些研究成果。具体如下：.（1）利用复双曲等距群中双曲元素的性质，给出了复Fuchs群的特征；.（2）建立了Mobius群、复双曲等距子群的几个离散性准则和代数收敛性定理;.（3）研究了复双曲群中椭圆元素的collar结构，给出了复双曲分支流形体积的一个估计。.（4）得到了几个无穷维离散Mobius子群中的几个不等式。