球面CR几何是三维球面在CR变换群PU（2,1）作用下的不变几何，本项目的主要目的是研究一些三维双曲流形（如：Whitehead 链环的补、 Gieseking 流形、类似于八字扭结补的流形等）的球面CR几何。具体地说，给定流形的一个理想三角剖分，我们将通过利用粘合CR四面体的方法来研究三维双曲流形的基本群到PU（2,1）中的表示；分析这些表示的离散性；讨论球面CR结构的存在性。另外，对于已得到的离散表示，我们将从离散群作用于复双曲空间的角度研究三维双曲流形的球面CR几何，即通过构造离散群的基本域，考虑其不连续集在群作用下的商空间的拓扑。

Spherical CR geometry is the PU(2,1) invariant geometry of the 3-sphere, where PU(2,1) is the CR transformation group. The main purpose of this program is to study the spherical CR geometry of some hyperbolic 3-manifolds, such as the complement of the Whitehead link, the Gieseking manifold and the sister of the complement of the figure eight knot. In detail, giving an ideal triangulation of the manifolds, we will study the representations from the fundamental group of the manifold to PU(2,1) by a geometric method of gluing CR tetrahedra, analysis the discreteness of the representations and discuss if there exist associate spherical CR structures. On the other hand, for the given discrete representation, we will study the spherical CR geometry of the manifold from the view of discrete groups acting on the complex hyperbolic space, which means that to construct the fundamental domain of the discrete group and then consider the topology of the quotient space of the discontinues set up to the group action.