Intrinsic Circle Domains

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain Ω in a compact Riemann surface S. This means that each connected component B of S \ Ω is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface (Ω ∪ B). Moreover, the pair (Ω, S) is unique up to conformal isomorphisms. We give a generalization to countably infinite connectivity. Finally, we show how one can compute numerical approximations to intrinsic circle domains using circle packings and conformal welding.