Quasiconformal Jordan Domains

We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y,dY). We say that a metric space (Y,dY) is a quasiconformal Jordan domain if the completion Y of (Y,dY) has finite Hausdor 2-measure, the boundary ∂Y=Y\Y is homeomorphic to S1, and there exists a homeo-morphism φ:D→(Y,dY) that is quasiconformal in the geometric sense. We show that φ has a continuous, monotone, and surjective extension Φ:D→Y. This result is best possible in this generality. In addition, we find a necessary and suffcient condition for Φ to be a quasiconformal homeomorphism. We provide suffcient conditions for the restriction of Φ to S1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.