On volumes of hyperideal tetrahedra with constrained edge lengths

TL;DR

This paper proves that regular hyperideal tetrahedra maximize volume among tetrahedra with constrained edge lengths below a certain threshold, aiding in hyperbolic 3-manifold volume computations.

Contribution

It establishes a volume maximization property for hyperideal tetrahedra with bounded edge lengths, advancing understanding of their geometric behavior.

Findings

Regular hyperideal tetrahedra maximize volume for edge lengths below a specific constant.

The result aids in computing the ideal simplicial volume of hyperbolic 3-manifolds.

Provides a fundamental step in geometric analysis of hyperbolic manifolds.

Abstract

Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces. The Schl\"afli formula neatly describes the behaviour of the volume of hyperideal tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove that, for every $\ell<\ell_0$, where $\ell_0$ is an explicit constant, regular hyperideal tetrahedra of edge length $\ell$ maximize the volume among hyperideal tetrahedra whose edge lengths are all not smaller than $\ell$. This result provides a fundamental step in the computation of the ideal simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic boundary.