Quasiconformal Jordan domains

英文摘要：

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 $ϕ\colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $ϕ$ has a continuous, monotone, and surjective extension $Φ\colon \overline{ \mathbb{D} } \rightarrow \overline{ Y }$. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for $Φ$ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of $Φ$ to $\mathbb{S}^{1}$ being a quasisymmetry and to $\partial Y$ being bi-Lipschitz equivalent to a quasicircle in the plane.