Combinatorial Ricci flow on cusped 3-manifolds

TL;DR

This paper introduces an extended combinatorial Ricci flow method for cusped 3-manifolds, providing a new way to find and characterize complete hyperbolic metrics through flow convergence.

Contribution

It extends the combinatorial Ricci flow to handle singularities, establishing an equivalence between flow convergence and the existence of hyperbolic metrics.

Findings

Flow extension through singularities is effective.

Convergence of the flow characterizes hyperbolic metric existence.

Provides an algorithm for hyperbolic metric computation.

Abstract

Combinatorial Ricci flow on a cusped $3$-manifold is an analogue of Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact $3$-manifolds with boundary for finding complete hyperbolic metrics on cusped $3$-manifolds. Dual to Casson and Rivin's program of maximizing the volume of angle structures, combinatorial Ricci flow finds the complete hyperbolic metric on a cusped $3$-manifold by minimizing the co-volume of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may develop singularities. We overcome this difficulty by extending the flow through the potential singularities using Luo-Yang's extension. It is shown that the existence of a complete hyperbolic metric on a cusped $3$-manifold is equivalent to the convergence of the extended combinatorial Ricci flow, which gives a new characterization of existence of a complete hyperbolic metric on a cusped $3$-manifold dual to Casson and Rivin's program. The extended combinatorial Ricci flow also provides an effective algorithm for finding complete hyperbolic metrics on cusped $3$-manifolds.