Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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R. Frigerio - M. Moraschini

On volumes of hyperideal tetrahedra with constrained edge lengths

created by moraschini on 02 Jan 2020


Published Paper

Inserted: 2 jan 2020
Last Updated: 2 jan 2020

Journal: Periodica Mathematica Hungarica
Year: 2019

ArXiv: 1801.05326 PDF


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.

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