On volumes of truncated tetrahedra with constrained edge lengths

Truncated 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 Lobachevsky–Schläfli formula neatly describes the behaviour of the volume of truncated 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 ℓ< ℓ0, where ℓ0 is an explicit constant, the regular truncated tetrahedron of edge length ℓ maximizes the volume among truncated tetrahedra whose edge lengths are all not smaller than ℓ. 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.