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

The aim of this paper is to study finite convex polyhedra in three dimensional hyperbolic space H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}^3$$\end{document}. 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 angle0≤Θ<π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \Theta < \pi $$\end{document}. That is, for any circle pattern on C^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mathbb {C}}$$\end{document}, its quasiconformal deformation space can be naturally identified with the product of the Teichmüller spaces of its interstices.