Geometric properties of φ -uniform domains

Abstract

We consider proper subdomains G of Rn and their images G′= f(G) under quasiconformal mappings f of Rn. We compare the distance ratio metrics of G and G′; as an application we show that φ-uniform domains are preserved under quasiconformal mappings of Rn. A sufficient condition for φ-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If G⊂ Rn is φ-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar φ-uniform domain whose complement is not ψ-uniform for any ψ. © 2016, Forum D'Analystes, Chennai.