Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds

TL;DR

This paper uses combinatorial Ricci flow techniques to establish conditions for the existence and uniqueness of cusped hyperbolic structures on 3-manifolds, demonstrating convergence properties of the flow.

Contribution

It introduces an extended Ricci flow method for decorated hyperbolic polyhedral metrics, proving long-time existence, uniqueness, and convergence criteria for hyperbolic structures on 3-manifolds.

Findings

Extended Ricci flow exists long-term and is unique for decorated hyperbolic metrics.

Flow converges exponentially to zero Ricci curvature metrics when they exist.

Results apply to cusped hyperbolic structures via ideal triangulation.

Abstract

In this paper, we adopt combinatorial Ricci curvature flow methods to study the existence of cusped hyperbolic structure on 3-manifolds with torus boundary. For general pseudo 3-manifolds, we prove the long-time existence and the uniqueness for the extended Ricci flow for decorated hyperbolic polyhedral metrics. We prove that the extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature. If it is the case, the flow converges exponentially fast. These results apply for cusped hyperbolic structure on 3-manifolds via ideal triangulation.