Characterizations of circle patterns and finite convex polyhedra in hyperbolic 3-space

Abstract

The aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space \({\mathbb {H}}^3\). We characterize the quasiconformal deformation space of each finite convex polyhedron. As a corollary, we obtain some results on finite circle patterns in the Riemann sphere with dihedral angle\(0\le \Theta < \pi \). That is, for any circle pattern on \(\hat{\mathbb {C}}\), its quasiconformal deformation space can be naturally identified with the product of the Teichmuller spaces of its interstices.