Combinatorial Ricci Flow and Thurston's Triangulation Conjecture

TL;DR

This paper introduces combinatorial Ricci flow as a novel method to approach Thurston's triangulation conjecture, establishing conditions for convergence and demonstrating the uniqueness of hyperbolic polyhedral structures.

Contribution

It provides a systematic approach using Ricci flow to verify Thurston's conjecture and proves the rigidity of hyperbolic polyhedral 3-manifolds with a focus on convergence criteria.

Findings

Ricci flow converges iff the triangulation is geometric

Hyperbolic polyhedral metrics are uniquely determined by cone angles

Flow evolution leads to complete hyperbolic structures

Abstract

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric decomposition into ideal hyperbolic tetrahedra, a result proven only for certain special 3-manifolds. This paper presents combinatorial Ricci flow as a systematic and general approach to addressing Thurston's triangulation conjecture, showing that the flow converges if and only if the triangulation is geometric. First, we prove the rigidity of the most general hyperbolic polyhedral 3-manifolds constructed by isometrically gluing partially truncated and decorated hyperbolic tetrahedra, demonstrating that the metrics are uniquely determined by cone angles modulo isometry and decoration changes. Then, we demonstrate that combinatorial Ricci flow evolves polyhedral metrics toward complete hyperbolic structures with geometric decompositions when convergent. Conversely, the existence of a geometric triangulation guarantees flow convergence.