Blaschke product for bordered surfaces

Abstract

Any ramified holomorphic covering of a closed unit disc by another such a disc is given by a finite Blaschke product. The inverse is also true. In this note we give two explicit constructions for a holomorphic ramified covering of a disc by other bordered Riemann surface. The machinery used here strongly resembles the description of magnetic configurations in submicron planar magnets.

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References

  1. 1.

    Achiezer, N.I.: Elements of Theory of Elliptic Functions. Nauka, Moscow (1970)

    Google Scholar 

  2. 2.

    Ahlfors, L.V.: Bounded analytic functions. Duke Math. J. 14(1), 1–11 (1947)

    Google Scholar 

  3. 3.

    Ahlfors, L.: Open Riemann surfaces and extremal problems on compact subregions. Comment. Math. Helv. 24, 100–134 (1950)

    Google Scholar 

  4. 4.

    Alling, N.L., Greenleaf, N.: Foundations of the Theory of Klein Surfaces. Springer, Berlin (1971)

    Google Scholar 

  5. 5.

    Bell, S.R.: The role of the Ahlfors mapping in the theory of kernel functions in the plane. In: Saitoh, S., Alpay, D., Ball, J.A., Ohsawa, T. (eds.) Reproducing Kernels and Their Applications (Newark, DE, 1997). International Society for Analysis, Applications and Computation, vol. 3, pp. 33–42. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  6. 6.

    Bogatyrev, A.B.: Real meromorphic differentials: a language for describing meron configurations in planar magnetic nanoelements. Theor. Math. Phys. 193(1), 1547–1559 (2017)

    Google Scholar 

  7. 7.

    Farkas, H.M., Kra, I.: Riemann Surfaces. Springer, Berlin (1980)

    Google Scholar 

  8. 8.

    Fay, J.: Theta Functions on Riemann Surfaces. Princeton University Press, Princeton (1974)

    Google Scholar 

  9. 9.

    Garabedian, P.R.: Schwarz’s lemma and the Szegö kernel function. Trans. Am. Math. Soc. 67(1), 1–35 (1949)

    Google Scholar 

  10. 10.

    Goluzin, G.M.: Geometric Function Theory of Functions of a Complex Variable. Nauka, Moscow (1952, 1966)

  11. 11.

    Grunsky, H.: Lectures on Theory of Functions in Multiply Connected Domains. Vandenhoeck & Ruprecht, Göttingen (1978)

    Google Scholar 

  12. 12.

    Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry, vols. 1 and 2. Wiley, New York (1994)

  13. 13.

    Khavinson, D.: On removal of periods of conjugate functions in multiply connected domains. Michigan Math. J. 31, 371–379 (1984)

    Google Scholar 

  14. 14.

    Khavinson, S.Y.: A method for removing the multivalence of analytic functions. Soviet Math. (Iz. VUZ) 34(11), 80–90 (1990)

    Google Scholar 

  15. 15.

    Kruzhilin, N.: Proper holomorphic maps of Reinhardt domains. In: Mergelyan-90 Memorial Conference, Yerevan (2018)

  16. 16.

    Mumford, D.: Tata Lectures on Theta. Springer, Berlin (1984)

    Google Scholar 

  17. 17.

    Orevkov, S.Y.: Separating semigroup of hyperelliptic curves and of genus 3 curves. arXiv:1712.03851

  18. 18.

    Schiffer, M., Spencer, D.C.: Functionals of Finite Riemann Surfaces. Princeton University Press, Princeton (1954)

    Google Scholar 

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Acknowledgements

Funding was provided by RAS Program “Fundamental mathematics and its applications” (Grant No. PRAS-18-01).

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Correspondence to A. B. Bogatyrev.

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To the memory of Sasha Vasil’ev.

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Supported by the Program of the Russ. Acad. Sci. ‘Fundamental Mathematics and its Applications’ under Grant PRAS-18-01.

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Bogatyrev, A.B. Blaschke product for bordered surfaces. Anal.Math.Phys. 9, 1877–1886 (2019). https://doi.org/10.1007/s13324-019-00284-z

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Keywords

  • Ahlfors map
  • Schottky double
  • Riemann bilinear relations
  • Theta functions

Mathematics Subject Classification

  • 30F50
  • 30F30
  • 30C70