Circle Patterns on Singular Surfaces

We consider “hyperideal” circle patterns, i.e. patterns of disks which do not cover the whole surface, which are associated to hyperideal hyperbolic polyhedra. The main result is that, on a Euclidean or hyperbolic surface with conical singularities, those hyperideal circle patterns are uniquely determined by the intersection angles of the circles and the singular curvatures. This is related to results on the dihedral angles of ideal or hyperideal hyperbolic polyhedra. The results presented here extend those in [Sch05a], however the proof is completely different (and more intricate) since [Sch05a] used a shortcut which is not available here.