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

TL;DR

This paper demonstrates that a specific combinatorial Ricci flow approach can be used to uniquely determine hyperbolic metrics with totally geodesic boundaries on certain compact 3-manifolds with ideal triangulations, ensuring exponential convergence.

Contribution

It introduces a novel application of a modified combinatorial Ricci flow to establish hyperbolization of compact 3-manifolds with boundary and ideal triangulations, proving exponential convergence.

Findings

Existence and uniqueness of hyperbolic metrics under given conditions.

Exponential convergence of the extended Ricci flow to the hyperbolic metric.

Applicability to 3-manifolds with ideal triangulations of valence at least 10.

Abstract

We prove that for a compact 3-manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that T is isotopic to a geometric decomposition of M. Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo [Luo05] for pseudo 3-manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.