Archiv der Mathematik

, Volume 106, Issue 1, pp 73–79 | Cite as

On behaviour of holomorphically contractible systems under non-monotonic sequences of sets

  • Arkadiusz Lewandowski
Open Access


The new results concerning the continuity of holomorphically contractible systems treated as set functions with respect to non-monotonic sequences of sets are given. In particular, continuity properties of Kobayashi and Carathéodory pseudodistances, as well as Lempert and Green functions with respect to sequences of domains converging in the Hausdorff metric are delivered.


Invariant pseudodistances Kobayashi pseudodistance Carathéodory pseudodistance Pluricomplex Green function Hausdorff distance 

Mathematics Subject Classification

Primary 32F45 Secondary 32H02 


  1. 1.
    Budzyńska M., Kuczumow T., Reich S., Limiting behavior of the Kobayashi distance, Taiwanese J. Math. 19, 535–552 (2015)MathSciNetGoogle Scholar
  2. 2.
    M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, 2nd edition, de Gruyter Expositions in Mathematrics 9, Walter de Gruyter 2014.Google Scholar
  3. 3.
    Kobayashi M., Convergence of Invariant Distances on Decreasing Domains, Complex Variables 47, 155–165 (2002)CrossRefMATHGoogle Scholar
  4. 4.
    Nivoche S., The pluricomplex Green function, capacitative notions, and approximation problems in \({\mathbb{C}^{n}}\), Indiana Univ. Math. J. 44, 489–510 (1995)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Van Trao N., Hue Minh T., Remark on the Kobayashi hyperbolicity of complex spaces, Acta Mathematica Vietnamica 34, 375–387 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer Science, Institute of MathematicsJagiellonian UniversityKrakówPoland
  2. 2.Science Institute, School of Engineering and Natural SciencesUniversity of IcelandReykjavíkIceland

Personalised recommendations