The Distortion Theorem for quasiconformal mappings, Schottky’s Theorem and holomorphic motions

Abstract

We prove the equivalence of Schottky’s theorem and the distortion theorem for planar quasiconformal mappings via the theory of holomorphic motions. The ideas lead to new methods in the study of distortion theorems for quasiconformal mappings and a new proof of Teichmüller’s distortion theorem.