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Combinatorial Ricci Flow and Thurston's Triangulation Conjecture

Feng Ke, Ge Huabin

TL;DR

This work presents a comprehensive program to realize Thurston's geometric triangulation conjecture through a generalized combinatorial Ricci flow on decorated hyperbolic tetrahedra. It proves a global rigidity result for decorated hyperbolic polyhedral metrics glued from multiple tetrahedron types and develops an extended CRF whose potential is a convex H-function; convergence of the flow to a zero-curvature decorated metric corresponds to the existence of a complete hyperbolic structure with a geometric triangulation. The main contributions include the Fundamental Theorem of the Combinatorial Ricci Flow, a rigorous extension of dihedral-angle data to degenerate configurations, and the demonstration that CRF (possibly with surgeries) provides a principled route to discovering geometric triangulations and proving hyperbolization in broad classes. The results establish a rigorous energy-minimization framework that ties geometric structure to topological conclusions, offering a new pathway toward Thurston's triangulation conjecture and a constructive approach to hyperbolic 3-manifolds.

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.

Combinatorial Ricci Flow and Thurston's Triangulation Conjecture

TL;DR

This work presents a comprehensive program to realize Thurston's geometric triangulation conjecture through a generalized combinatorial Ricci flow on decorated hyperbolic tetrahedra. It proves a global rigidity result for decorated hyperbolic polyhedral metrics glued from multiple tetrahedron types and develops an extended CRF whose potential is a convex H-function; convergence of the flow to a zero-curvature decorated metric corresponds to the existence of a complete hyperbolic structure with a geometric triangulation. The main contributions include the Fundamental Theorem of the Combinatorial Ricci Flow, a rigorous extension of dihedral-angle data to degenerate configurations, and the demonstration that CRF (possibly with surgeries) provides a principled route to discovering geometric triangulations and proving hyperbolization in broad classes. The results establish a rigorous energy-minimization framework that ties geometric structure to topological conclusions, offering a new pathway toward Thurston's triangulation conjecture and a constructive approach to hyperbolic 3-manifolds.

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.

Paper Structure

This paper contains 21 sections, 31 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The geometry of hyperbolic tetrahedra
  3. Five types of hyperbolic tetrahedra
  4. Geometry of $2$-$2$ type hyperbolic tetrahedra
  5. Geometry of $3$-$1$ type hyperbolic tetrahedra
  6. No lower dimensional degeneracy
  7. The extension of the dihedral angles of hyperbolic tetrahedra
  8. $\phi$-functions and the decorated edge length space ${\mathcal{R}}$
  9. The geometric structure of ${\mathcal{L}}$ and ${\mathcal{R}}$
  10. Extending $\alpha$ from ${\mathcal{L}}$ to $\overline{{\mathcal{R}}}$ continuously
  11. The volume and co-volume function of hyperbolic tetrahedra
  12. The volume and co-volume function
  13. The extended co-volume function
  14. Global rigidity of hyperbolic polyhedral 3-manifolds
  15. Decorated polyhedral metric and combinatorial Ricci curvature
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose $(M,{\mathcal{T}})$ is a triangulated compact pseudo 3-manifold, then a decorated hyperbolic polyhedral metric on $(M,{\mathcal{T}})$ is determined up to isometry and change of decorations by its curvature.

Figures (2)

  • Figure 1: decorated edge lengthes in the Poincaré disk model
  • Figure 2: a $2$-$2$ type hyperbolic tetrahedron

Theorems & Definitions (64)

  • Theorem 1.1: Rigidity
  • Theorem 1.2: Fundamental Theorem of the Combinatorial Ricci Flow
  • Corollary 1.3
  • Conjecture 1
  • Remark 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 54 more