Quasiconformal mirrors
Let \(L\) be a Jordan curve on the Riemann sphere \(\widehat{\mathbb C} = {\mathbb C} \cup \{ \infty \}\). According to Ahlfors, an orientation-reversing quasiconformal mapping of \(\widehat{\mathbb C}\) onto itself which keeps every point of \(L\) fixed is said to be a quasiconformal reflection with respect to \(L\). Define the quasiconformal dilatation \(Q_L\) of a set \(L\) by \(Q_L=(1+q_L)/(1-q_L)\), with \(q_L = \inf \{ k(f) : f\) is a quasiconformal reflection with respect to \(L\}\), \(k(f)\) is the dilatation of \(f\). Define, analogously, a quasiconformal reflection with respect to an arbitrary set \(E\subset \widehat{\mathbb C}\) as an orientation-reversing quasiconformal mapping of \(\widehat{\mathbb C}\) which keeps every point of \(E\) fixed. The quasiconformal dilatation of \(E\) can also be defined in a similar manner. In [Complex analysis, Artic. dedicated to Albert Pfluger, 139-156 (1988; Zbl 0659.30016)], \textit{R. Kühnau} conjectured that a set \(E \subset \widehat{\mathbb C}\) admits a quasiconformal reflection if and only if it is a subset of some quasiconformal circle. The main result of the article asserts that every subset \(E \subset \widehat{\mathbb C}\) with \(q_E < 1\) is a subset of some quasiconformal circle \(L\) whose quasiconformal dilatation satisfies the inequality \(Q_L \leq Q_E^4\). This gives an affirmative answer to the Kühnau conjecture and provides other important corollaries.
scientific article; zbMATH DE number 1396797