Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range

Abstract

We study the minimum-volume problem for finite-volume convex hyperbolic polyhedra whose dihedral angles are all equal to a fixed number \(\alpha\). In the non-obtuse case one necessarily has \[ \frac{\pi}{3}\le \alpha\le \frac{\pi}{2}. \] We prove that throughout the full range in which the regular hyperbolic tetrahedron with dihedral angle \(\alpha\) exists, \[ \frac{\pi}{3}\le \alpha<\arccos\frac13, \] it is the unique minimum-volume equiangular hyperbolic polyhedron with prescribed angle \(\alpha\). The left endpoint is the known ideal case, while at \(\alpha=\arccos(1/3)\) the regular tetrahedron degenerates to the Euclidean one. The proof combines Andreev's theorem and the Schl\"afli formula with Atkinson's decomposition into atoroidal and prismatic parts, explicit volume estimates for ordinary prisms and complete orthoschemes, and a direct equiangular version of Inoue's edge surgery.