Milin's coefficients, complex geometry of Teichmüller spaces and variational calculus for univalent functions

Abstract

Abstract We investigate the invariant metrics and complex geodesics in the universal Teichmüller space and the Teichmüller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all non-expanding invariant metrics in either of these spaces coincide with its intrinsic Teichmüller metric. Other applications concern the variational theory for univalent functions with quasiconformal extension. It turns out that geometric features caused by the equality of metrics and connection with complex geodesics provide deep distortion results for various classes of such functions and create new phenomena which do not appear in the classical geometric function theory.