The Möbius midpoint condition as a test for quasiconformality and the quasimöbius property

Abstract

The Möbius midpoint condition, introduced by Goldberg in 1974 as a criterion for the quasisymmetry of a mapping of the line onto itself and considered by Aseev and Kuzin in 1998 in the same role for the topological embeddings of the line into ℝⁿ, yields no information on the quasiconformality or quasisymmetry of a topological embedding of ℝ^k into ℝⁿ for 1 < k ≤ n. In this article we introduce a Möbius-invariant modification of the midpoint condition, which we call the “Möbius midpoint condition” MMC(f) ≤ H < 1. We prove that if this condition is fulfilled then every homeomorphism of domains in $\overline {\mathbb{R}^n }$ is K(H)-quasiconformal, while a topological embedding of the sphere $\overline {\mathbb{R}^k }$ into $\overline {\mathbb{R}^n }$ (for 1 ≤ k ≤ n) is ω _H-quasimöbius. The quasiconformality coefficient of K(H) and the distortion function ω _H depend only on H and are expressed by explicit formulas showing that K(H) → 1 and ω _H → id as H → 1/2. Since MMC(f) = 1/2 is equivalent to the Möbius property of f, the resulting formulas yield the closeness of the mapping to a Möbius mapping for H near 1/2.