The Journal of Analysis

, Volume 24, Issue 1, pp 57–66 | Cite as

Geometric properties of \(\varphi \)-uniform domains

  • Peter Hästö
  • Riku KlénEmail author
  • Swadesh Kumar Sahoo
  • Matti Vuorinen
Original Research Paper


We consider proper subdomains G of \({\mathbb R}^n\) and their images \(G'=f(G)\) under quasiconformal mappings f of \({\mathbb R}^n\). We compare the distance ratio metrics of G and \(G'\); as an application we show that \(\varphi \)-uniform domains are preserved under quasiconformal mappings of \({\mathbb R}^n\). A sufficient condition for \(\varphi \)-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If \(G\subset {\mathbb R}^n\) is \(\varphi \)-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar \(\varphi \)-uniform domain whose complement is not \(\psi \)-uniform for any \(\psi \).


The distance ratio metric The quasihyperbolic metric Uniform and \(\varphi \)-uniform domains John domains Quasiconformal and quasisymmetric mappings 

Mathematics Subject Classification

Primary 30F45; Secondary 30C65 30L05 30L10 


  1. Gehring, F.W. 1999. Characterizations of quasidisks, vol. 48., Quasiconformal geometry and dynamics Warszawa: Banach Center Publications, Polish Academy of Sciences.zbMATHGoogle Scholar
  2. Gehring, F.W., and K. Hag. 2012. The ubiquitous quasidisk. With contributions by Ole Jacob Broch. Mathematical Surveys and Monographs, vol. 184., American Mathematical Society, Providence, RI, 2012Google Scholar
  3. Gehring, F.W., K. Hag, and O. Martio. 1989. Quasihyperbolic geodesics in John domains. Mathematica Scandinavica 65: 75–92.MathSciNetCrossRefzbMATHGoogle Scholar
  4. Gehring, F.W., and B.G. Osgood. 1979. Uniform domains and the quasihyperbolic metric. Journal d'Analyse Mathematique 36: 50–74.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Gehring, F.W., and B.P. Palka. 1976. Quasiconformally homogeneous domains. Journal d'Analyse Mathematique 30: 172–199.MathSciNetCrossRefzbMATHGoogle Scholar
  6. Herron, D.A. 1999. John domains and the quasihyperbolic metric. Complex Variables 39: 327–334.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Kim, K., and N. Langmeyer. 1998. Harmonic measure and hyperbolic distance in John disks. Mathematica Scandinavica 83: 283–299.MathSciNetCrossRefzbMATHGoogle Scholar
  8. Klén, R., Y. Li, S.K. Sahoo, and M. Vuorinen. 2014. On the stability of \(\varphi \)-uniform domains. Monatshefte für Mathematik 174(2): 231–258.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Koskela, P. 2009. Lectures on quasiconformal and quasisymmetric mappings. Lecture note, December 2009.Google Scholar
  10. Martio, O., and J. Sarvas. 1979. Injectivity theorems in plane and space. Annales Academiae Scientiarum Fennicae Mathematica 4: 384–401.MathSciNetzbMATHGoogle Scholar
  11. Näkki, R., and J. Väisälä. 1991. John disks. Expositiones Mathematicae 9: 3–43.MathSciNetzbMATHGoogle Scholar
  12. Tukia, P., and J. Väisälä. 1980. Quasisymmetric embeddings of metric spaces. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 5(1): 97–114.Google Scholar
  13. Väisälä, J. 1971. Lectures On \(n\) -Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics 229, Springer, New York.Google Scholar
  14. Väisälä, J. 1988. Uniform domains. Tohoku Mathematical Journal 40: 101–118.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Väisälä, J. 1991. Free quasiconformality in Banach spaces II. Annales Academiae Scientiarum Fennicae Mathematica 16: 255–310.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Väisälä, J. 1998. Relatively and inner uniform domains. Conformal Geometry and Dynamics 2: 56–88.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Vuorinen, M. 1985. Conformal invariants and quasiregular mappings. Journal d'Analyse Mathematique 45: 69–115.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Vuorinen, M. 1988. Conformal geometry and quasiregular mappings, vol. 1319. Lecture Notes in Mathematics, Berlin: Springer.Google Scholar

Copyright information

© Forum D'Analystes, Chennai 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of OuluOuluFinland
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland
  3. 3.Discipline of MathematicsIndian Institute of Technology IndoreIndoreIndia

Personalised recommendations