On the stability of $\varphi$-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 $\phi$-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\phi$ from $[0,\infty )$ to itself. Finally, we discuss a number of stability properties of $\phi$-uniform domains. In particular, we show that the class of $\phi$-uniform domains is stable in the sense that removal of a geometric sequence of points from a $\phi$-uniform domain yields a ${\phi }_1$-uniform domain.

23 pages, 7 figures

Cited by in corpus (2)

• Geometry of Metrics
• Volume growth of quasihyperbolic balls