Tests for extremality in a problem for quasiconformal mappings
The author considers an extremal problem posed by Belinskij. Consider the class of quasiconformal (qc) mappings from a Riemann surface R to a Riemann surface S which are homotopic to a given quasiconformal mapping f from R to S and which have maximal dilatation Q. Assume the minimal dilatation of mappings homotopic modulo the ideal boundary to f is K(f) and that \(Q>K(f)\). Let \(z_ 1,...,z_ n\) be distinct points of R and let \(F(w_ 1,...,w_ n)\) be a real-valued differentiable function with domain the n-fold product of the surface S and assume F has nonvanishing gradient. One considers the problem (of Belinskij) of finding the maximal value of \(F(f)=F(f(z_ 1),f(z_ 2),...,f(z_ n))\) as f ranges over all qc mappings in the same homotopy class with dilatation Q. The author gives a necessary and sufficient condition for a function \(f_ 0\) to be extremal for this problem. The function \(f_ 0^{-1}\) must satisfy a certain supremum condition relative to quadratic differentials which are holomorphic except for certain simple poles.
scientific article; zbMATH DE number 3966628