On the stability of φ-uniform domains

Abstract

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain G ⊂ ℝn. In the sequel, we investigate a class of domains, so called φ-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism φ from [0,∞) to itself. Finally, we discuss a number of stability properties of φ-uniform domains. In particular, we show that the class of φ-uniform domains is stable in the sense that removal of a geometric sequence of points from a φ-uniform domain yields a φ1-uniform domain. © 2013 Springer-Verlag Wien.