Quasicircles and the conformal group (Q5890782)

From the introduction: ``The 2-sphere \( {\mathbb S}^2 \), oriented and equipped with its standard conformal structure, is isomorphic to the complex projective line \(P^1 {\mathbb C} \simeq {\mathbb C }\cup \{ \infty\}\). Let \( K \geq 1\). By definition, a \(K\)-quasicircle \(c \subset {\mathbb S}^2 \) is the image \( c = f(c_0) \) of a circle \( c_0 \subset {\mathbb S}^2 \) under a \(K\)-quasiconformal homeomorphism \(f : {\mathbb S}^2 \rightarrow {\mathbb S}^2\). The aim in this paper is to characterize those Jordan curves in \( {\mathbb S}^2\) that are quasicircles in terms of their orbit under the action of the conformal group G := Conf \(^+({\mathbb S}^2) \simeq \mathrm{PSl}_2{\mathbb C} \).'' NEWLINENEWLINEThe authors prove that a Jordan curve in the 2-sphere is a quasicircle if and only if the closure of its orbit under the action of the conformal group contains only points and Jordan curves.NEWLINENEWLINEEditorial remark: This version of the article was retracted by the authors in September 2016 because the findings had been previously published. It is identical to Zbl 1357.30013.