The Related Topics Of Curve Complex And Heegaard Splittings

3-manifolds are always an important topic in low dimensional topology. For studying the properties of 3-manifolds, Heegaard introduced the Heegaard splittings for closed orientable 3-manifolds and Haken extended the Heegaard splittings to compact orientable cases.The curve complex, as a main tool in studying Heegaard splittings, has been known by lots of mathematicians. Masur-Minsky gave the curve complex a metric and proved that the curve complex with this metric is hyperbolic. After that, many papers appeared to give this result some different proofs. By using the points of R3, we give the curve complex some different metrics and study the properties of the curve complex with these metrics. Our conclusion maybe help us to give a new proof about the hyperbolicity of the curve complex.To understand the Heegaard splittings of 3-manifolds, Hempel introduced the Heegaard distance for a Heegaard splitting and he proved that there are 3-manifolds with distance arbitrar-ily large Heegaard splittings. After that, lots of mathematicians have given lots of results about this topic. Inspired by their work, we prove that:for any positive integer number g? 2, there are infinitely many 3-manifolds each of which has a genus g and distance 2 Heegaard splitting. Our conclusion reveals the general distribution of hyperbolic 3-manifolds.By Thurston's Geometrization Conjecture, hyperbolic 3-manifolds are an important object in studying 3-manifolds. Since there is always a triangulation for any compact orientable 3-manifold, it is an important question that what triangulation of a 3-manifold corresponds to the hyperbolicity. To study this question, Casson introduced the angle structure in the ideal trian-gulation of 3-manifolds, and he proved that if a 3-manifold with torial boundary has an ideal triangulation which admits the angle structure, then the interior of the 3-manifold is hyperbolic. On the other hand, Hodgson-Rubinstein-Segerman proved that there is an ideal triangulation which admits the angle structure on a cusped hyperbolic 3-manifold which homeomorphic to the interior of a compact 3-manifold with torus boundary components if the topology of this 3-manifold satisfy some conditions. So a natural question is what is the relationship between the triangulation and the hyperbolicity of 3-manifolds with large genus boundaries. In the cases of 3-manifolds with large genus boundaries, Luo also introduced the angle structure as Casson's. Using Kojima's work of the ideal polyhedra decomposition in a hyperbolic 3-manifold with to-tally geodesic boundaries, we prove that any hyperbolic 3-manifold with large genus boundaries admits an ideal triangulation that supports angle structures. On the other hand, we also prove that if a 3-manifold with totally geodesic boundaries admits an ideal triangulation that supports angle structures, then the 3-manifold is hyperbolic.