Circle Patterns on Singular Surfaces

Abstract

We consider “hyperideal” circle patterns, i.e., patterns of disks appearing in the definition of the weighted Delaunay decomposition associated with a set of disjoint disks, possibly with cone singularities at the centers of those disks. Hyperideal circle patterns are associated with hyperideal hyperbolic polyhedra. We describe the possible intersection angles and singular curvatures of those circle patterns on Euclidean or hyperbolic surfaces with cone singularities. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra. The results presented here extend those in Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]), however, the proof is completely different (and more intricate) since Schlenker (Math. Res. Lett. 12(1), 82–112, [2005]) used a shortcut which is not available here.