Combinatorial Ricci flows and the hyperbolization of a class of compact 3–manifolds

摘要

We prove that for a compact $3$--manifold $M$ with boundary admitting an ideal triangulation $mathcal{T}$ with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that $mathcal{T}$ is isotopic to a geometric decomposition of $M!$. Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo (href{http://dx.doi.org/10.1090/S1079-6762-05-00142-3}{Electron. Res. Announc. Amer. Math. Soc. 11 (2005) 12--20}) for pseudo-$3$--manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.