The smallest hyperbolic 3-manifolds with totally geodesic boundary (Q806069)

The authors prove that, among compact hyperbolic 3-manifolds with non- empty totally geodesic boundary, each one having the minimum volume admits a polyhedral decomposition by two regular truncated tetrahedra of dihedral angle \(\pi\) /6. Moreover, using a definite integral of some elementary functions, they show that this minimal volume (it is twice the volume of a regular truncated tetrahedron of dihedral angle \(\pi\) /6) is 6.452.... For a classification of such manifolds with the minimal volume, see \textit{M. Fujii} [Hyperbolic 3-manifolds with totally geodesic boundary which are decomposed into hyperbolic truncated tetrahedra, Tokyo J. Math. 13, No.2, 353-373 (1990; Zbl 0729.57005)].