Finite Coverings in the Hyperbolic Plane

Abstract

We prove that if a (th r)-convex domain in the hyperbolic plane is covered by n ≥ 2 circular discs of radius r, then the density of the covering is larger than 2π/ \sqrt{27}. The density bound is optimal, and the condition of (th r)-convexity is essentially optimal. Combining our result with earlier estimates yields that if at least two non-overlapping equal circular discs cover a given circular disc in a surface of constant curvature, then the density of the covering is larger than 2π/ \sqrt{27}.