三维流形上的双曲几何

在本项目中,我们主要进行广义的双曲几何及其在三维流形中的应用等相关问题的研究. 其中我们这里的几何包括有限体积双曲流形,无限体积双曲流形(既Kleinian 群的形变理论),各种与曲面相关的组合复形的Gromov 双曲几何, 以及双曲的多面体几何.而关于三维流形的理论我们这里关心三维流形的拓扑、几何,以及它们之间的关系. 特别是,我们关心三维流形的Heegaard 分解这一拓扑结构与三维流形的双曲几何的关系. 我们研究的具体问题是：1，skinning 映射与Heegaard 分解的距离的关系; 2，柄体的多面体双曲几何; 3, Riley 片的几何极限,以及一般的柄体上的双曲结构的几何极限; 4，柄体的双曲结构的一致几何模型及在闭双曲三维流形上的应用; 5,Hatcher-Thurston 复形的大范围几何及在Heegaard 分解中的应用.

We will study hyperbolic geometry and 3-manifolds. Here, by hyperbolic geometry, we mean finite volume hyperbolic 3-manifolds, infinite volume hyperbolic 3-manifolds (i.e., the deformation theory of Kleinian groups), the hyperbolic geometry in the sense of Gromov of combinatorial complexes related to a surface,and hyperbolic polyhedral geometry. We are interested in the relationship between the topologies and the geometries of 3-manifolds. In particular, we will study the relation between Heegaard splittings of 3-manifolds and hyperbolic geometry. More precisely,we will study the following problems:1,the relation between skinning map and Heegaard distance; 2,hyperbolic polyhedral geometry of handlebodies; 3,geometrical limits of Riley groups, and geometrical limits of general hyperbolic structures on handlebodies; 4, uniform geometrical models of hyperbolic handlebodies and its application on closed hyperbolic 3-manifolds;5, the large scale geometry of Hatcher-Thurston complex and its application on Heegaard splittings.