由 Thurston 的几何化猜想(已被证明)和 Hempel 的一些结果可知, 任意一个有距离不小于 3 的 Heegaard 分解的三维流形是双曲三维流形. 因此 Heegaard 距离能够给出双曲三维流形的一个初步刻画. 但是一些结果表明总是存在无穷多个不同胚的双曲三维流形有给定距离, 给定亏格的 Heegaard 分解. 从而仅仅从 Heegaard 距离出发, 实现双曲三维流形的分类是不可能的.. Zhang-Qiu-Zou 通过研究曲线复形上不同类型的边, 构造了一系列新的度量. 本项目拟研究如何利用这些度量来给出 Heegaard 分解的一些新指标, 从而能够解读出双曲三维流形更多的几何拓扑性质. 此外本项目也将研究环面边界双曲三维流形上角度结构的存在性问题, 并试图给出一些新的结果.

By Thurston’s Geometric Conjecture and some results of Hempel, every 3-manifold with a Heegaard distance at least 3 Heegaard splitting is hyperbolic. So there is a description of hyperbolic 3-manifolds by the Heegaard distance. But, some results show that there are infinitely many non-homeomorphic hyperbolic 3-manifolds with the same distance and genus Heegaard splittings. So it is impossible to give a completely classification of hyperbolic 3-manifolds along the Heegaard distance. . By the types of edges, Zhang-Qiu-Zou constructed a series of new metrics on the curve complex. In this project, we will study how to use these new metrics to give some new indexes of Heegaard splittings so as to find more characterizations of hyperbolic 3-manifolds. Moreover, we will also study the existence of angle structure in a hyperbolic 3-manifold with torus boundary and try to give some results.