Generalizing Andreev's Theorem via circle patterns

TL;DR

This paper extends the Circle Pattern Theorem to include obtuse angles, enabling a broader characterization of hyperbolic polyhedra and generalizing Andreev's Theorem through a novel discrete regularity framework.

Contribution

It introduces an extended Circle Pattern Theorem for obtuse angles and generalizes Andreev's Theorem for hyperbolic polyhedra with new proof techniques.

Findings

Extended circle pattern theorem for obtuse angles

Characterization of hyperbolic polyhedra with obtuse dihedral angles

New discrete regularity scheme for proofs

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.