Generalizing Andreev's Theorem via circle patterns

Abstract

In this paper we derive an extended Circle Pattern Theorem that allows obtuse overlap angles. As a consequence, we characterize a subclass of compact convex hyperbolic polyhedra with possibly obtuse dihedral angles and thus generalize Andreev's Theorem. For proofs of these results, we establish a discrete analog of the "weak solution/regularity theory" scheme.

27 pages, 4 figures