Quasiconformal Jordan domains

Abstract

We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains $(Y,d_Y)$. We say that a metric space $(Y,d_Y)$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $(Y,d_Y)$ has finite Hausdorff $2$-measure, the boundary $\partial Y=\overline{Y}\setminus Y$ is homeomorphic to ${\mathbb{S}}^1$, and there exists a homeomorphism $\varphi :\mathbb{D}\to (Y,d_Y)$ that is quasiconformal in the geometric sense. We show that $\varphi$ has a continuous, monotone, and surjective extension $\Phi :\overline{\mathbb{D}}\to \overline{Y}$. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for $\Phi$ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of $\Phi$ to ${\mathbb{S}}^1$ being a quasisymmetry and to $\partial Y$ being bi-Lipschitz equivalent to a quasicircle in the plane.

arXiv comment21 pages; revised version