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

TL;DR

This paper investigates the rigidity and volume optimization of 1-3 type hyperbolic polyhedral 3-manifolds, demonstrating that their metrics are uniquely determined by curvature and extending volume optimization results to these structures.

Contribution

It introduces new rigidity results for decorated 1-3 type hyperbolic polyhedral metrics and extends Casson-Rivin's volume optimization program to these manifolds.

Findings

Decorated 1-3 type hyperbolic polyhedral metrics are determined by curvature.

Casson-Rivin's volume optimization applies to 1-3 type ideal triangulated 3-manifolds.

Existence of an angle structure implies strongly 1-efficient triangulation.

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.