Quasiconformal Jordan domains

TL;DR

This paper extends the Carathéodory extension theorem to quasiconformal Jordan domains in metric spaces, establishing conditions for continuous extensions and characterizing when these extensions are quasiconformal or quasisymmetric.

Contribution

It generalizes classical conformal extension results to quasiconformal Jordan domains in metric spaces, providing new criteria for extensions to be quasiconformal or quasisymmetric.

Findings

Extension $orall$ quasiconformal Jordan domains is continuous, monotone, and surjective.

Characterization of when the extension is a quasiconformal homeomorphism.

Conditions under which the boundary is bi-Lipschitz to a quasicircle.

Abstract

We extend the classical Carath\'eodory 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 $\phi \colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $\phi$ has a continuous, monotone, and surjective extension $\Phi \colon \overline{ \mathbb{D} } \rightarrow \overline{ Y }$. 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.