Monatshefte für Mathematik

, Volume 174, Issue 2, pp 231–258 | Cite as

On the stability of \(\varphi \)-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 \(\varphi \)-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.


The quasihyperbolic metric The distance ratio metric \(j\) Uniform domains \(\varphi \)-uniform domains Quasiconvex domains Removability 

Mathematics Subject Classification (1991)

Primary 30F45 Secondary 30C65 


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© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  2. 2.Department of MathematicsHunan Normal UniversityChangshaPeople’s Republic of China
  3. 3.Department of MathematicsIndian Institute of Technology IndoreIndoreIndia

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