Combinatorial Ricci flow on compact 3-manifolds with boundary

TL;DR

This paper proves Luo's conjecture that combinatorial Ricci flow can algorithmically find complete hyperbolic metrics on compact 3-manifolds with boundary by extending the flow through singularities.

Contribution

It extends the combinatorial Ricci flow to handle singularities, confirming its effectiveness in computing hyperbolic metrics on 3-manifolds with boundary.

Findings

Proves Luo's conjecture affirmatively.

Extends Ricci flow through singularities.

Establishes algorithmic approach for hyperbolic metrics.

Abstract

Combinatorial Ricci flow on an ideally triangulated compact 3-manifold with boundary was introduced by Luo as a 3-dimensional analog of Chow-Luo's combinatorial Ricci flow on a triangulated surface and conjectured to find algorithmically the complete hyperbolic metric on the compact 3-manifold with totally geodesic boundary. In this paper, we prove Luo's conjecture affirmatively by extending the combinatorial Ricci flow through the singularities of the flow if the ideally triangulated compact 3-manifold with boundary admits such a metric.