学术报告（葛化彬 12.14）

After the Hyperbolization Theorem, the existence of hyperbolic structure is clear.However, for a hyperbolic 3-manifold, the existence of a geometric triangulation is still open. My talk is along the direction of solving this conjecture.

Using combinatorial Ricci flow methods, we show: Let M be a compact 3-manifold with boundary consisting of surfaces of genus at least 2. If M admits an ideal triangulation with valence at least 10 at all edges, then there exists a unique hyperbolic metric on M with totally geodesic boundary under which the ideal triangulation is geometric. This provides the first existence result of a geometric triangulation on such 3-manifolds, and shows a deep connection between the topology and the geometry of 3-manifolds. Moreover, the combinatorial Ricci flow provides an effective tool of finding geometric structures and geometric triangulations of 3-manifolds. The talk is based on joint work with Ke Feng and Bobo Hua.