Necessary and sufficient conditions for extremality in certain classes of quasiconformal mappings
The author gives necessary and sufficient conditions for quasiconformal mappings to be extremal in certain classes. To define the class one starts with a Fuchsian group \(\Gamma\) acting on the upper half plane U. \(L_{\infty}(\Gamma)\) is the Banach space of Beltrami differentials \(\nu\) satisfying \(\nu\) (\(\gamma\) (w))\({\bar \gamma}\)'(w)/\(\gamma\) '(w)\(=\nu (w)\) for every \(\gamma\) in \(\Gamma\). For \(\nu\) in \(L_{\infty}(\Gamma)\) with \(\| \nu \|_{\infty}<1\), \(F_{\nu}(w)\) is the uniquely determined quasiconformal self-mapping of \(\bar U\) with complex dilatation \(\nu\) (w) and normalized at 0,1,\(\infty\). \(\sigma\) is a T-invariant closed subset of the real line \({\hat {\mathbb{R}}}\). E is a T-invariat measurable subset of U such that \(D=U\setminus E\) has positive measure. b(w) is a nonnegative bounded measurable function on E, automorphic for \(\Gamma\), satisfying \(\| b\|_{\infty}=es\sup b(w)<1\). \(h: {\hat {\mathbb{R}}}\to {\hat {\mathbb{R}}}\) is the boundary mapping included by some \(F_{\nu}.\) The class \(Q=Q(\Gamma,h,\sigma,E,b)\) consists of those \(F_{\nu}\) with \(\| \nu \|_{\infty}<1\), \(\nu\) in \(L_{\infty}(\Gamma)\), which satisfy \(F_{\nu}|_{\sigma}=h| \sigma\) and \(| \nu (w)| \leq b(w)\) a.e. in E. This paper gives necessary and sufficient conditions for a mapping to be extremal in Q provided that the space of integrable holomorphic quadraic differentials, real off of \(\sigma\), is finite dimensional or provided that E/\(\Gamma\) is relatively compact in \(\{\) \(U\cup ({\hat {\mathbb{R}}}-\sigma)\}/\Gamma\).
scientific article; zbMATH DE number 3975308