Skip to main content
SpringerLink
Log in

Uniformization of one-parametric families of complex tori

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We suggest an approximate method to find an elliptic function uniformizing a compact Riemann surface of genus 1 which is given as a ramified covering of the Riemann sphere. The method is based on including the surface into a smooth one-parametric family. We deduce a system of ordinary differential equations for critical points of elliptic functions uniformizing surfaces of the family.

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. Aleksandrov, I. A. Parametric Continuations in the Theory of Univalent Functions (Nauka, Moscow, 1976) [in Russian].

    MATH  Google Scholar 

  2. Goluzin, G. M. Geometric Theory of Functions of a Complex Variable (Nauka, Moscow, 1966; Transl. of Math. Monographs, Vol. 26, AMS, Rhode Island, 1969).

    MATH  Google Scholar 

  3. Gutlyanskii, V. Ya. and Ryazanov, V. I. Geometric and Topological Theory of Functions and Mappings, Zadachi i Metody. Matematika,Mekhanika, Kibernetika 5 (Naukova Dumka, Kyïv, 2011.) [in Russian].

    Google Scholar 

  4. Branges, L. de “A Proof of the Bieberbach Conjecture”, ActaMathematica 154, No. 1, 137–152 (1985).

    MathSciNet  MATH  Google Scholar 

  5. Goryainov, V. V. “Semigroups of Analytic Functions in Analysis and Applications”, Russian Mathematical Surveys 67, No. 6, 975–1021 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. Bauer, M., Bernard, D., Kytola, K. “Multiple Schramm–Loewner Evolutions and Statistical Mechanics Martingales”, J. Stat. Phys. 120, No. 5–6, 1125–1163 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  7. Graham, K. “On Multiple Schramm–Loewner Evolutions”, Journal of Statistical Mechanics: Theory and Experiment, No. 3 (2007).

  8. Dubedat, J. “Commutation Relations for Schramm–Loewner Evolutions”, Comm. Pure Appl. Math. 60, No. 12, 1792–1847 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  9. Nasyrov, S. R. “Determination of the Polynomial Uniformizing a Given Compact Riemann Surface”, Mathematical Notes, 91, No. 4, 558–567, (2012).

    Article  MathSciNet  MATH  Google Scholar 

  10. Nasyrov, S. R. “One-Parametric Families of Elliptic Functions Uniformizing Complex Tori”, in Proc. of VII Petrozavodsk Int. Conf. ‘Complex Analysis and Its Appl.’ (Petrozavodsk State Univ., Petrozavodsk, 2014), pp. 78–79.

    Google Scholar 

  11. Nasyrov, S. R. “One-Parametric Families of Multivalent Funcions and Riemann Surfaces”, in Proc. of Int. Voronezh Winter Math. School ‘Modern Methods in Function Theory and Related Topics’ (Voronezh State Univ., Voronezh, 2015), pp. 83–85.

    Google Scholar 

  12. Hurwitz, A. “Über Riemannsche Flächen mit gegeben Verzweigungpunkten”, Math. Ann. 39, 1–61 (1891).

    Article  MathSciNet  Google Scholar 

  13. Hurwitz, A. “Über die Anzahl der Riemannsche Flächen mit gegeben Verzweigungpunkten”, Math. Ann. 55, 53–66 (1902).

    Article  Google Scholar 

  14. Weyl, H. “Über dasHurwitzscheProblemder Bestimmung der Anzahl Riemannscher Flächen von gegebener Verzweigungsart”, Comment.Math. Helv. 3, 103–113 (1931).

    Article  MathSciNet  MATH  Google Scholar 

  15. Lloyd, E. “Riemann Surface Transformation Groups”, J. Combinatorial Theory (A) 13, 17–27 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  16. Mednykh A.D. “Non-Equivalent Coverings of Riemann Surfaces With a Given Branch-Type”, Siberian Math. J. 25, No. 4, 120–142 (1984) [in Russian].

    Google Scholar 

  17. Mednykh, A. D. “A New Method for Counting Coverings Over Manifold With Finitely Generated Group”, DokladyMathematics 74, No. 1, 498–502 (2006).

    MATH  Google Scholar 

  18. Nasyrov, S. R. Geometric Problems in Theory of Ramified Coverings of Riemann Surfaces (Magarif, Kazan, 2008) [in Russian].

    Google Scholar 

  19. Lando, S. K. “Ramified Coverings of the Two-Dimensional Sphere and the Intersection Theory in Spaces of Meromorphic Functions on Algebraic Curves”, RussianMathematical Surveys 57, No. 3, 463–533 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  20. Lando, S. K. and Zvonkin, A. K. Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences (Springer Verlag, Berlin–Heidelberg–New York, 2004), Vol.141.

  21. Hurwitz, A. and Courant, R. Function Theory (Mir,Moscow, 1968) [Russian translation].

    Google Scholar 

  22. Akhiezer, N. I. Elements of the Theory of Elliptic Functions (Fizmatlit, Moscow, 1970; AMS Transl. of Math. Monographs, Vol. 79, AMS, Rhode Island, 1990).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kazan Federal University, ul. Kremlyovskaya 18, Kazan, 420008, Russia

    S. R. Nasyrov

Authors
  1. S. R. Nasyrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. R. Nasyrov.

Additional information

Original Russian Text © S.R. Nasyrov, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 8, pp. 42–52.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nasyrov, S.R. Uniformization of one-parametric families of complex tori. Russ Math. 61, 36–45 (2017). https://doi.org/10.3103/S1066369X17080047

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X17080047

Keywords

Search

Navigation