On the stability of $ϕ$-uniform domains

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 $φ$-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $φ$ from $[0,\infty)$ 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 $φ_1$-uniform domain.

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