Monatshefte für Mathematik

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

On the stability of \(\varphi \)-uniform domains

Article

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.

Keywords

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

Mathematics Subject Classification (1991)

Primary 30F45 Secondary 30C65 

References

  1. 1.
    Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.K.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)MATHGoogle Scholar
  2. 2.
    Gehring, F.W.: Characterizations of quasidisks. Quasiconformal Geom. Dyn. 48, 11–41 (1999)MathSciNetGoogle Scholar
  3. 3.
    Gehring, F.W., Osgood, B.G.: Uniform domains and the quasihyperbolic metric. J. Anal. Math. 36, 50–74 (1979)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Gehring, F.W., Palka, B.P.: Quasiconformally homogeneous domains. J. Anal. Math. 30, 172–199 (1976)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Hästö, P., Ponnusamy, S., Sahoo, S.K.: Inequalities and geometry of the Apollonian and related metrics. Rev. Roumaine Math. Pures Appl. 51, 433–452 (2006)MATHMathSciNetGoogle Scholar
  6. 6.
    Klén, R.: On hyperbolic type metrics, Dissertation, University of Turku, Turku. Ann. Acad. Sci. Fenn. Math. Diss. 152, 49 (2009)Google Scholar
  7. 7.
    Lindén, H.: Quasihyperbolic geodesics and uniformity in elementary domains, Dissertation, University of Helsinki, Helsinki, 2005. Ann. Acad. Sci. Fenn. Math. Diss. 146, 50 (2005)Google Scholar
  8. 8.
    MacManus, P.: The complement of a quasimöbius sphere is uniform. Ann. Acad. Sci. Fenn. Math. 21, 399–410 (1996)MathSciNetGoogle Scholar
  9. 9.
    Martin, G.J., Osgood, B.G.: The quasihyperbolic metric and the associated estimates on the hyperbolic metric. J. Anal. Math. 47, 37–53 (1986)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Math. 4, 384–401 (1979)MathSciNetGoogle Scholar
  11. 11.
    Väisälä, J.: Uniform domains. Tohoku Math. J. 40, 101–118 (1988)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Väisälä, J.: Free quasiconformality in Banach spaces II. Ann. Acad. Sci. Fenn. Math. 16, 255–310 (1991)MATHGoogle Scholar
  13. 13.
    Väisälä, J.: Relatively and inner uniform domains. Conform. Geom. Dyn. 2, 56–88 (1998)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Vuorinen, M.: Conformal invariants and quasiregular mappings. J. Anal. Math. 45, 69–115 (1985)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, vol. 1319. Springer, Berlin (1988)Google Scholar

Copyright information

© 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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