Quasiconformal Jordan Domains

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