Skip to main content
SpringerLink
Log in

On the boundary extension of mappings on Riemannian surfaces in terms of prime ends

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We investigate non-homeomorphic mappings of the Riemannian surfaces of Sobolev class. Some distortion estimates have been obtained for the moduli of families of paths. We have proved that under some conditions, these mappings have a continuous extension to the boundary of a domain in terms of prime ends.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. F. Beardon, The geometry of discrete groups. Graduate Texts in Math, vol. 91, New York, Springer-Verlag, 1983.

  2. C. Caratheodory, “¨ Uber die Begrenzung der einfachzusammenhängender Gebiete,” Mathematische Annalen, 73, 323–370 (1913).

    Article  MathSciNet  MATH  Google Scholar 

  3. B. Fuglede, “Extremal length and functional completion,” Acta Math., 98, 171–219 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Gol’dshtein and A. Ukhlov, “Traces of functions of L12 Dirichlet spaces on the Caratheéodory boundary,” Studia Math., 235(3), 209–224 (2016).

  5. V. Gutlyanskii, V. Ryazanov, and E. Yakubov, “The Beltrami equations and prime ends,” Ukr. Mat. Visn., 12(1), 27–66 (2015); English transl. in J. Math. Sci., 210(1), 22–51 (2015).

  6. J. Hesse, “A p−extremal length and p-capacity equality,” Ark. Mat., 13, 131–144 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  7. D. P. Ilyutko and E. A. Sevost’yanov, “Boundary behaviour of open discrete mappings on Riemannian manifolds,” Sb. Math., 209(5), 605–651 (2018).

  8. D. A. Kovtonyuk and V. I. Ryazanov, “Prime ends and Orlicz-Sobolev classes,” St. Petersburg Math. J., 27(5), 765–788 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Kuratowski, Topology, v. 2. New York–London, Academic Press, 1968.

    MATH  Google Scholar 

  10. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory. New York, Springer Science + Business Media, 2009.

    MATH  Google Scholar 

  11. V. M. Miklyukov, “Relative Lavrientiev distance and prime ends on non-parametric surfaces,” Ukr. Mat. Visnyk, 1(3), 349–372 (2004).

    MATH  Google Scholar 

  12. R. Näkki, “Boundary behavior of quasiconformal mappings in n-space,” Ann. Acad. Sci. Fenn. Ser. A., 484, 1–50 (1970).

    MathSciNet  MATH  Google Scholar 

  13. R. Näkki, “Extension of Loewner’s capacity theorem,” Trans. Amer. Math. Soc., 180, 229–236 (1973).

    MathSciNet  MATH  Google Scholar 

  14. R. Näkki, “Prime ends and quasiconformal mappings,” J. Anal. Math., 35, 13–40 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Ryazanov and S. Volkov, On the Boundary Behavior of Mappings in the Class W1,1loc on Riemann Surfaces. Complex Analysis and Operator Theory, 11, 1503–1520 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Ryazanov and S. Volkov, “Prime ends in the Sobolev mapping theory on Riemann surfaces,” Mat. Stud., 48, 24–36 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  17. R. R. Salimov and E. A. Sevost’yanov, “On the Equicontinuity of One Family of Inverse mappings in Terms of Prime Ends,” Ukr. Math. J., 70(9), 1456–1466 (2019).

  18. E. A. Sevost’yanov, “On the Boundary and Global Behavior of Mappings of Riemannian Surfaces,” Filomat, 36(4), 1295–1327 (2022).

  19. E. A. Sevost’yanov, “On boundary extension of mappings in metric spaces in terms of prime ends,” Ann. Acad. Sci. Fenn. Math., 44(1), 65–90 (2019).

  20. E. S. Smolovaya, “Boundary behavior of ring Q-homeomorphisms in metric spaces,” Ukr. Math. Journ., 62(5), 785–793 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  21. G. D. Suvorov, Generalized “principle of a length and area” in the mapping theory. Kiev, Naukova Dumka, 1985 (in Russian).

    MATH  Google Scholar 

  22. J. Väisälä, “Discrete open mappings on manifolds,” Ann. Acad. Sci. Fenn. Ser. A 1 Math., 392, 1–10 (1966).

  23. J. Väisälä, “Lectures on n-dimensional quasiconformal mappings,” Lecture Notes in Math., 229. Berlin etc., Springer-Verlag, 1971.

  24. S. V. Volkov and V. I. Ryazanov, “On the boundary behavior of the class W1,1loc on Riemannian surfaces,” Proc. Inst. Appl. Mech. of NASU, 29, 34–53 (2015).

    Google Scholar 

  25. W. P. Ziemer, “Extremal length and conformal capacity,” Trans. Amer. Math. Soc., 126(3), 460–473 (1967).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Zhytomyr Ivan Franko State University, Zhytomyr, Ukraine

    Evgeny Sevost’yanov, Oleksandr Dovhopiatyi & Nataliya Ilkevych

  2. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk, Ukraine

    Evgeny Sevost’yanov

Authors
  1. Evgeny Sevost’yanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleksandr Dovhopiatyi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nataliya Ilkevych
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vitalina Kalenska
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 20, No. 2, pp. 241–257, April–June, 2023

Presented by V. Gutlyanskyĭ

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sevost’yanov, E., Dovhopiatyi, O., Ilkevych, N. et al. On the boundary extension of mappings on Riemannian surfaces in terms of prime ends. J Math Sci 274, 370–382 (2023). https://doi.org/10.1007/s10958-023-06606-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06606-8

Keywords

Search

Navigation