On the stability of $\phi$-uniform domains

Abstract

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n$. In the sequel, we investigate a class of domains, so called $\varphi$-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\varphi$ from $[0,\infty)$ to itself. Finally, we discuss a number of stability properties of $\varphi$-uniform domains. In particular, we show that the class of $\varphi$-uniform domains is stable in the sense that removal of a geometric sequence of points from a $\varphi$-uniform domain yields a $\varphi_1$-uniform domain.