Quasiconformality of the injective mappings transforming spheres to quasispheres

Abstract

We prove that every injective mapping of a domain \(D \subset \overline {{\mathbb{R}^n}} \) transforming spheres Σ ⊂ D to K-quasispheres (the images of spheres under K-quasiconformal automorphisms of \(\overline {{\mathbb{R}^n}} \)) is K′-quasiconformal with K′ depending only on K and tending to 1 as K → 1. This is a quasiconformal analog of the classical Carathéodory Theorem on the Möbius property of an injective mapping of a domain D ⊂ Rⁿ which sends spheres to spheres.