Intrinsic circle domains

Abstract

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain Ω \Omega in a compact Riemann surface S S . This means that each connected component B B of S ∖ Ω S\setminus \Omega is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface ( Ω ∪ B ) (\Omega \cup B) . Moreover, the pair ( Ω , S ) (\Omega , 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.