Minimal Equiangular Hyperbolic Polyhedra in the Tetrahedral Range

Abstract

We consider finite-volume convex hyperbolic polyhedra whose dihedral angles are all equal to a fixed number $\alpha$. Such non-obtuse equiangular polyhedra may exist only for $\frac{\pi}{3}\leq\alpha\leq\frac{\pi}{2}$. The endpoint cases of the minimal-volume problem are known: for $\alpha=\frac{\pi}{3}$, the minimum is attained by the ideal regular tetrahedron, while for $\alpha=\frac{\pi}{2}$, among right-angled polyhedra, it is attained by the triangular bipyramid $P(3,2)$, whose volume is Catalan's constant. We prove the corresponding statement in the tetrahedral range $\frac{\pi}{3}\leq\alpha<\arccos\left(\frac{1}{3}\right)$, where the regular hyperbolic tetrahedron with dihedral angle $\alpha$ exists. Namely, for every such $\alpha$, among all equiangular hyperbolic polyhedra with dihedral angle $\alpha$, the minimum volume is attained only by this tetrahedron. The proof combines Andreev's theorem, the Schl\"afli formula, 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.