Rigidity, volume and angle structures of 1-3 type hyperbolic polyhedral 3-manifolds

TL;DR

The paper addresses rigidity and volume optimization for hyperbolic 3-manifolds built from decorated 1-3 type hyperbolic tetrahedra. It proves that decorated 1-3 type hyperbolic polyhedral metrics are determined by curvature up to isometry and decoration, and that Casson–Rivin’s volume optimization program extends to these manifolds; the co-volume is a locally convex potential whose gradient recovers edge dihedral data, enabling a Fenchel dual formulation. By extending dihedral angles and the co-volume to all of $\mathbb{R}^6$ and establishing strict convexity, the authors obtain global rigidity and a rigorous variational framework for angle structures, including existence and uniqueness of volume-maximizing angle structures and their realization by decorated edge lengths. The work unifies and extends prior approaches (Luo–Yang, Lackenby, Casson–Rivin) to the $1$-$3$ type setting, paving the way for hyperbolization results and efficient triangulations in broader classes of hyperbolic 3-manifolds.

Abstract

In this paper, we study the rigidity of hyperbolic polyhedral 3-manifolds and the volume optimization program of angle structures. We first study the rigidity of decorated 1-3 type hyperbolic polyhedral metrics on 3-manifolds which are isometric gluing of decorated 1-3 type hyperbolic tetrahedra. Here a 1-3 type hyperbolic tetrahedron is a truncated hyperbolic tetrahedron with one hyperideal vertex and three ideal vertices. A decorated 1-3 type polyhedron is a 1-3 type hyperbolic polyhedron with a horosphere centered at each ideal vertex. We show that a decorated 1-3 type hyperbolic polyhedral metric is determined up to isometry and change of decorations by its curvature. We also prove several results on the volume optimization program of Casson and Rivin, i,e. Casson-Rivin's volume optimization program is shown to be still valid for 1-3 type ideal triangulated 3-manifolds. We also get a strongly 1-efficiency triangulation when assuming the existence of an angle structure. On the whole, we follow the spirit of Luo-Yang's work in 2018 to prove our main results. The main differences come from that the hyperbolic tetrahedra considered here have completely different geometry with those considered in Luo-Yang's work in 2018.

Figures (8)

• Figure 1: Configuration of a 1-3 type hyperbolic tetrahedron
• Figure 2: left (right): a $1$-$3$ type (flat $1$-$3$ type) hyperbolic tetrahedron
• Figure 3: decorated edge lengths in Poincaré's disk model
• Figure 4: Notations in a $M_{1|3}$ tetrahedron
• Figure 5: Configuration of a quadrilateral

Theorems & Definitions (65)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Casson
• Theorem 1.4: Casson, Rivin
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1: Bao2002Fujii1990Schlenker2002
• Lemma 2.2
• proof
• Remark 2.1