Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact $3$-Manifolds with Boundary

In this paper, we study a combinatorial Ricci flow on closed pseudo $3$-manifolds $(M,\mathcal{T})$. We prove that if every edge in the triangulation $\mathcal{T}$ has valence at least $9$, then the combinatorial Ricci flow converges exponentially fast to a hyperbolic metric. As a consequence, for any compact $3$-manifold $N$ with boundary admitting an ideal triangulation ${\mathcal{T}}_N$ whose edges all have valence at least $9$, there exists a unique complete hyperbolic metric with totally geodesic boundary on $N$ such that ${\mathcal{T}}_N$ is isotopic to a geometric decomposition of $N$. This provides a partial solution to the conjecture of Costantino, Frigerio, Martelli and Petronio, and hence an affirmative answer of Thurston’s geometric ideal triangulation conjecture for such manifolds. Moreover, we obtain explicit upper and lower bounds for the resulting hyperbolic metric.