Quasiconformal Jordan Domains

Abstract

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 Hausdorff 2-measure, the boundary ∂Y = ̄Y \ Y is homeomorphic to 𝕊1, and there exists a homeomorphism 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 sufficient condition for Φ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of Φ to 𝕊1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.

Polyhedral approximation and uniformization for non-length surfaces

“…Moreover, combining Theorem 1.8 with the uniformization result of Theorem 1.2 or Theorem 1.5 and with a result of Ikonen [20], we also show that in order to obtain reciprocity at points of ∂X it suffices to verify condition (1.4) rather than upper reciprocity. Theorem 1.10.…”

“…We remark that without requiring any condition on ∂X, the reciprocity of int(X) does not imply the reciprocity of X in general; this was observed in [20,Sect. 1.1].…”

Two‐dimensional metric spheres from gluing hemispheres

“…As a related note, it is clear, for example, by [21, Theorem 1.1 and Proposition 1.2], that the inclusion map $$\widetilde{\iota }_{2}$$ is a 1‐quasiconformal homeomorphism if and only if there exists a quasiconformal homeomorphism $$h \colon \widetilde{\iota }_{2}(\overline{Z}_{2} ) \rightarrow \overline{ \mathbb {D} }$$, where $$\overline{ \mathbb {D} }$$ is the closed Euclidean unit disk.…”