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

Abstract

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 (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.