Skip to main content
SpringerLink
Log in

Asymptotic Conformal Welding via Löwner-Kufarev Evolution

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

The Löwner-Kufarev evolution produces asymptotics for mappings onto domains close to the unit disk \(\mathbb{D }\) or the exterior of \(\mathbb{D }\). We deduce variational formulae which lead to the asymptotic conformal welding for such domains. The comparison of mappings onto bounded and unbounded components of the Jordan curve establishes an asymptotic connection between driving functions in both versions of the Löwner-Kufarev equation and conformal radii of the two domains.

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. Bishop, C.J.: Conformal welding and Koebe’s theorem. Ann. Math. 166, 613–656 (2007)

    Article  MATH  Google Scholar 

  2. Contreras, M.D., Díaz-Madrigal, S., Gumenyuk, P.: Local duality in Loewner equation. Preprint 2012. arXiv: 1202.2334v1 [math.CV].

  3. Jones, G.L.: Conformal welding of Jordan curves using weighted Dirichlet spaces. Ann. Acad. Sci. Fenn. Math. 25(2), 405–412 (2000)

    MathSciNet  Google Scholar 

  4. Kufarev, P.P.: On one-parameter families of analytic functions. Rec. Math. [Mat. Sbornik] N.S. 13(55), 87–118 (1943)

  5. Lavrentyev, M.A., Shabat, B.V.: Methods of the Theory of Functions of a Complex Variable. Nauka, Moscow (1973)

    Google Scholar 

  6. Lebedev, N.A.: Application of the area principle to the problems for nonoverlapping domains. Tr. Math. Inst. Acad. Sci. USSR 60, 211–231 (1961)

    MATH  Google Scholar 

  7. Lebedev, N.A.: The Area Principle in the Theory of Univalent Functions. Nauka, Moscow (1975)

    MATH  Google Scholar 

  8. Löwner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89(1–2), 103–121 (1923)

    Article  MathSciNet  MATH  Google Scholar 

  9. Markina, I., Vasil’ev, A.: Virasoro algebra and dynamics in the space of univalent functions. Contemp. Math. 525, 85–116 (2010)

    Article  MathSciNet  Google Scholar 

  10. Pommerenke, Ch.: Über die Subordination analytischer Funktionen. J. Reine Angew. Math. 218, 159–173 (1965)

    Google Scholar 

  11. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  12. Prokhorov, D.: Conformal welding for domains close to a disk. Anal. Math. Phys. 1(3), 101–114 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Siryk, G.V.: On a conformal mapping of near domains. Uspekhi Matem. Nauk 9(5), 57–60 (1956)

    MathSciNet  Google Scholar 

  14. Takhtajan, L.A., Teo, L.-P.: Weil-Petersson Metric on the Universal Teichmüller Space. Mem. Am. Math. Soc. 183(861) (2006)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Mechanics, Saratov State University, Saratov, 410012, Russia

    Dmitri Prokhorov

Authors
  1. Dmitri Prokhorov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitri Prokhorov.

Additional information

Communicated by Lawrence Zalcman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prokhorov, D. Asymptotic Conformal Welding via Löwner-Kufarev Evolution. Comput. Methods Funct. Theory 13, 37–46 (2013). https://doi.org/10.1007/s40315-012-0002-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40315-012-0002-y

Keywords

Mathematics Subject Classification (1991)

Search

Navigation