A note on complete hyperbolic structures on ideal triangulated 3-manifolds

TL;DR

This paper extends Casson and Rivin's theorem by showing that the volume-maximizing property of the complete hyperbolic metric on ideal triangulated 3-manifolds remains valid even when some tetrahedra are flat, broadening the understanding of geometric structures.

Contribution

It demonstrates that the volume maximization result applies to cases with flat tetrahedra, generalizing previous theorems on hyperbolic structures.

Findings

Complete hyperbolic metric maximizes volume even with flat tetrahedra.

The volume maximization principle extends beyond strictly positive angle structures.

The result broadens the class of ideal triangulated 3-manifolds with known volume-maximizing metrics.

Abstract

It is a theorem of Casson and Rivin that the complete hyperbolic metric on a cusp end ideal triangulated 3-manifold maximizes volume in the space of all positive angle structures. We show that the conclusion still holds if some of the tetrahedra in the complete metric are flat.