On the stability of $φ$-uniform domains

Details On the stability of $φ$-uniform domains On the stability of $φ$-uniform domains

2008-12-23

arxiv.org/abs/0812.4369v4

Mathematics

Metric Geometry

31 pages, 11 figures

Scientific paper

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n$ and prove several inequalities between them. Next we study a class of domains, so called $\phi$-uniform domains, defined by the property that these two metrics are comparable. Finally we show that this class of 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.

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