On extremal quasiconformal mappings compatible with a Fuchsian group with a dilatation bound
Let U be the upper half-plane, \(\Gamma\) a Fuchsian group, \(\sigma\) a \(\Gamma\)-invariant subset of the extended real line \({\hat {\mathbb{R}}}\), which contains the limit set of \(\Gamma\), which, in turn, contains the points 0, 1 and \(\infty\). Let E be a \(\Gamma\) invariant subset of U such that the closure of E/\(\Gamma\) in (U\(\cup ({\hat {\mathbb{R}}}- \sigma))/\Gamma\) is a compact proper subset of \((U\cup ({\hat {\mathbb{R}}}- \sigma))/\Gamma.\) Let b(w) be a nonnegative bounded measurable function on E which is automorphic for \(\Gamma\) and which satisfies \(c_ 1= \sup_{w\in E}b(w)<1.\) Let M(\(\Gamma)\) be the set of Beltrami differentials for \(\Gamma\) with support in U and let \(z=F_{\nu}(w)\) be the unique quasiconformal automorphism of U which has Beltrami coefficient \(\nu\) and which leaves 0, 1 and \(\infty\) fixed. Within M(\(\Gamma)\) is the class \(M_{\mu}=M_{\mu}(\Gamma,\sigma,E,b)\) consisting of those \(\nu\) in M(\(\Gamma)\) for which \(F_{\nu}|_{\sigma}=F_{\mu}|_{\sigma}\) and for which \(| \nu (w)| \leq b(w).\) This article provides interesting sufficient conditions for \(\mu\) to be extremal within the class \(\mu\). The conditions are also necessary with mild assumptions about ess inf b(w). The method makes use of a variational lemma about the existence of trivial curves of Beltrami differentials which vanish on a given set.
scientific article; zbMATH DE number 3902987