Quasicircles, quasidisks, quasiarcs, and quasilines

Assume the Axiom of Choice. Write $\widehat{\mathbb{C}}=\mathbb{C}\cup \{\mathrm{\infty }\}$ for the Riemann sphere with its standard holomorphic charts (The Riemann sphere is the published one-point compactification of the complex plane, The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity). Put $\mathbb{T}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ and $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$. Identify the quotient circle ${\mathbb{S}}^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ 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 in $\widehat{\mathbb{C}}$ is the image of a topological embedding ${\mathbb{S}}^1\hookrightarrow \widehat{\mathbb{C}}$ (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological); by Jordan–Brouwer separation it has exactly two complementary components with common boundary. A sphere homeomorphism is $K$-quasiconformal if it preserves the standard complex orientation (Orientation-preserving homeomorphisms and the geometric definition of quasiconformality) and, in every connected holomorphic chart neighborhood, its coordinate expression is $K$-quasiconformal in the analytic sense (The ACL and Sobolev analytic definition of quasiconformality); $K\ge 1$ is a uniform upper bound for the local dilatations.

(a) A Jordan curve $\mathrm{\Gamma }\subseteq \widehat{\mathbb{C}}$ is a $K$-quasicircle if $\mathrm{\Gamma }=F(\mathbb{T})$ for a $K$-quasiconformal sphere homeomorphism $F$. It is a quasicircle if it is a $K$-quasicircle for some finite $K$. Its quasicircle constant is $K(\mathrm{\Gamma })=\inf \{K\ge 1:\mathrm{\Gamma }=F(\mathbb{T})\text{ for some }K\text{-quasiconformal sphere homeomorphism }F\}\mathrm{.}$ The infimum is not asserted to be attained.

(b) A domain $U\subseteq \widehat{\mathbb{C}}$ is a $K$-quasidisk if $U=G(\mathbb{D})$ for a $K$-quasiconformal sphere homeomorphism $G$; it is a quasidisk if this holds for some finite $K$. The two complementary components of a $K$-quasicircle are $K$-quasidisks.

(c) A $K$-quasiarc is the image of the open line segment $(-1,1)\subset \mathbb{C}$ under a $K$-quasiconformal homeomorphism of $\mathbb{C}$; a quasiarc is a $K$-quasiarc for some finite $K$. A $K$-quasiline is the image of $\mathbb{R}$ under a $K$-quasiconformal homeomorphism of $\mathbb{C}$; a quasiline is a $K$-quasiline for some finite $K$.

(d) Quasicircles are Möbius invariant with unchanged constant: for every Möbius transformation $M$, $K(M(\mathrm{\Gamma }))=K(\mathrm{\Gamma })\mathrm{.}$