A quasiconformal dynamic property of the disk (Q1385422)

Let \(D\) be a hyperbolic simply-connected domain in \(\widehat{\mathbb{C}}\), \(z_0\in D\) and \(0<r<1\). The \(r\)-circle of \(D\) with centre \(z_0\) is \(\gamma(z_0,r)= f^{-1} \{z:| z|= \frac 1r\}\) where \(f\) is a conformal map of \(D\) onto the disc \(\Delta= \{z:| z|>1\), \(z\in \widehat{\mathbb{C}}\}\) such that \(f(z_0)= \infty\). \(D\) is said to have the \(r^2\)-property if for every \(r\), \(0<r<1\) and every \(z_0\in D\) and every conformal map \(g\) of \(D\) into \(\widehat{\mathbb{C}}\) there is an \(r^2\)-quasiconformal map \(h:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}\) whose restriction to the interior of \(\gamma(z_0,r)\) is \(g\). \textit{S. L. Krushkal} [Sib. Math. J. 30, 730-744 (1989; Zbl 0701.30019)] proved that any disc has the \(r^2\)-property. The authors give two proofs of the converse result that among the hyperbolic simply-connected domains only discs have the \(r^2\)-property.