The welding homeomorphism of a Jordan curve

Assume the Axiom of Choice. Let $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$, $\mathbb{T}=\mathrm{\partial }\mathbb{D}$, and ${\mathbb{D}}^{\ast }=\widehat{\mathbb{C}}\setminus \bar{\mathbb{D}}$. Identify ${\mathbb{S}}^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ with $\mathbb{T}$ by $[t]\mapsto e^{2\pi it}$ (The circle as $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ with basepoint $[0]$, $[t]\mapsto (\cos 2\pi t,\sin 2\pi t)$ is a homeomorphism from $\mathbb{R}\mathrm{/}\mathbb{Z}$ to the unit circle). A Jordan curve is used in the sense of Quasicircles, quasidisks, quasiarcs, and quasilines; it has two complementary components with common boundary by Jordan–Brouwer separation. Fix an ordered pair of these components, denoted ${\mathrm{\Omega }}_0,{\mathrm{\Omega }}_1$.

The boundary-correspondence lemma (Riemann maps of Jordan domains extend to homeomorphisms of the closures) supplies conformal equivalences $f:\mathbb{D}\to {\mathrm{\Omega }}_0$ and $g:{\mathbb{D}}^{\ast }\to {\mathrm{\Omega }}_1$ and unique homeomorphic extensions $\bar{f}:\bar{\mathbb{D}}\to \overline{{\mathrm{\Omega }}_0}$ and $\bar{g}:\overline{{\mathbb{D}}^{\ast }}\to \overline{{\mathrm{\Omega }}_1}$. Here maps between spherical domains are conformal in the holomorphic charts of the Riemann sphere (The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity). Define the welding homeomorphism of $\mathrm{\Gamma }$ for the chosen maps by $h_{\mathrm{\Gamma };f,g}:=(\bar{f}{\mathrm{\mid }}_{\mathbb{T}}{)}^{-1}\circ (\bar{g}{\mathrm{\mid }}_{\mathbb{T}}):\mathbb{T}\to \mathbb{T}\mathrm{.}$

The map $h_{\mathrm{\Gamma };f,g}$ is orientation-preserving. An oriented conformal welding of an orientation-preserving homeomorphism $h:\mathbb{T}\to \mathbb{T}$ is a triple $(\mathrm{\Gamma },f,g)$ as above for which $h_{\mathrm{\Gamma };f,g}=h$. Equivalently, $\bar{g}{\mathrm{\mid }}_{\mathbb{T}}=\bar{f}{\mathrm{\mid }}_{\mathbb{T}}\circ h$. The convention $h=f^{-1}\circ g$ on the boundary is the inverse of the source convention $g^{-1}\circ f$.

The dependence on the parameter maps is a two-sided $\mathrm{Aut}(\mathbb{D})$ action. Let $J(z)=1\mathrm{/}z$, which maps ${\mathbb{D}}^{\ast }$ biholomorphically to $\mathbb{D}$. For $\alpha ,\beta \in \mathrm{Aut}(\mathbb{D})$, put ${\beta }^{\ast }:=J^{-1}\circ \beta \circ J\in \mathrm{Aut}({\mathbb{D}}^{\ast })$. Replacing $f$ by $f\circ \alpha$ and $g$ by $g\circ {\beta }^{\ast }$ changes the welding map to $h_{\mathrm{\Gamma };f\circ \alpha ,g\circ {\beta }^{\ast }}={\alpha }^{-1}{\mathrm{\mid }}_{\mathbb{T}}\circ h_{\mathrm{\Gamma };f,g}\circ {\beta }^{\ast }{\mathrm{\mid }}_{\mathbb{T}}\mathrm{.}$

Postcomposing both parameter maps with a Möbius transformation does not change the welding map: if $M$ is Möbius, the maps $M\circ f,M\circ g$ parameterize the correspondingly ordered components of $M(\mathrm{\Gamma })$ and $h_{M(\mathrm{\Gamma });M\circ f,M\circ g}=h_{\mathrm{\Gamma };f,g}$. No uniqueness of the welding curve is asserted here.