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Numerical Schottky Uniformizations: Myrberg’s Opening Process

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2013)

Abstract

It is well known that a closed Riemann surface may be described by different king of objects; for instance, by algebraic curves, Fuchsian groups, Schottky groups, Riemann period matrices, ext. In general, if one knows explicitly one of these presentations, it is a very hard problem to provide the others in an explicit way. In the 1920s, Myrberg [Myr16] proposed an algorithm which permits to approximate numerically a Schottky uniformization of a hyperelliptic Riemann surface once an explicit hyperelliptic curve presentation is given.

Keywords

  • Riemann Surface
  • Kleinian Group
  • Fuchsian Group
  • Riemann Sphere
  • Hyperelliptic Curve

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ahlfors, L., Sario, L.: Riemann Surfaces. Princeton Mathematical Series 26. Princeton University Press, Princeton, N.J. (1960)

    Google Scholar 

  2. Ahlfors, L.: Finitely generated Kleinian groups. Am. J. Math. 86, 413–429 (1964)

    CrossRef  MathSciNet  Google Scholar 

  3. Belokolos, E.D., Bobenko, A.I., Enol’ski, V.Z., Its, A.R., Matveev, V.B.: Algebro-geometric approach to nonlinear integrable equations. Springer Series in Nonlinear Dynamics. Springer, Berlin (1994)

    MATH  Google Scholar 

  4. Bers, L.: Automorphic forms for Schottky groups. Adv. Math. 16, 332–361 (1975)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Bers, L.: On the Ahlfors’ finiteness theorem. Am. Math. J. 89 (4), 1078–1082 (1967)

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Bobenko, A.I.: Schottky uniformization and finite-gap integration, Soviet Math. Dokl. 36, No. 1, 38–42 (1988) (transl. from Russian: Dokl. Akad. Nauk SSSR, 295, No. 2 (1987))

    Google Scholar 

  7. Burnside, W.: Note on the equation y 2 = x(x 4 − 1). Proc. Lond. Math. Soc. (1) 24, 17–20 (1893)

    Google Scholar 

  8. Burnside, W.: On a class of Automorphic Functions. Proc. Lond. Math. Soc. 23, 49–88 (1892)

    CrossRef  Google Scholar 

  9. Buser, P., Silhol, R.: Geodesic, Periods and Equations of Real Hyperelliptic Curves. Duke Math. J. 108, 211–250 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Chuckrow, V.: On Schottky groups with applications to Kleinian groups. Ann. Math. 88, 47–61 (1968)

    CrossRef  MathSciNet  Google Scholar 

  11. Farkas, H., Kra, I.: Riemann Surfaces. Second edition. Graduate Texts in Mathematics 71. Springer, New York (1992)

    Google Scholar 

  12. Gianni, P., Seppälä, M., Silhol, R., Trager, B.: Riemann surfaces, plane algebraic curves and their period matrices. J. Symbolic Comput. 26, 789–803 (1998)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Hidalgo, R.A., Maskit, B.: On neoclassical Schottky groups. Trans. AMS. 358, 4765–4792 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Hidalgo, R.A., Figueroa, J.: Numerical Schottky uniformizations. Geometriae Dedicata 111, 125–157 (2005)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Keen, L.: On Hyperelliptic Schottky groups. Ann. Acad. Sci. Fenn. Ser. A.I. Math. 5 (1), 165–174 (1980)

    Google Scholar 

  16. Klein, F.: Neue Beiträge zur Riemann’schen Funktionentheorie. Math. Ann. 21, 141–218 (1883)

    CrossRef  MathSciNet  Google Scholar 

  17. Koebe, P.: Über die Uniformisierung der Algebraischen Kurven II. Math. Ann. 69, 1–81 (1910)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Lebedev, Y.: OpenMath Library for Computing on Riemann Surfaces. Ph. D. Thesis, Florida State University (2008). http://etd.lib.fsu.edu/theses/available/etd-11102008-184256/unrestricted/LebedevYDissertation.pdf

  19. Lehto, O.: Univalent Functions and Teichmüller Spaces. GTM, vol. 109, Springer, New York (1986)

    Google Scholar 

  20. Marden, A.: Schottky groups and circles. In: Contributions to Analysis, pp. 273–278. Academic, New York and London (1974)

    Google Scholar 

  21. Maskit, B.: Remarks on m-symmetric Riemann surfaces. Contemp. Math. 211, 433–445 (1997)

    MathSciNet  Google Scholar 

  22. McMullen, C.: Complex Dynamics and Renormalization. Annals of Mathematical Studies, vol. 135. Princeton University Press, Princeton (1984)

    Google Scholar 

  23. Myrberg, J.P.: Über die Numerische Ausführung der Uniformisierung. Acta Soc. Scie. Fenn. XLVIII(7), 1–53 (1920)

    Google Scholar 

  24. Semmler, K.-D. and Seppälä, M.: Numerical uniformization of hyperelliptic curves. In: Proceedings of ISSAC 1995 (1995)

    Google Scholar 

  25. Seppälä, M.: Myrberg’s numerical uniformization of hyperelliptic curves. Ann. Acad. Scie. Fenn. Math. 29, 3–20 (2004)

    MATH  Google Scholar 

  26. Seppälä, M.: Computation of period matrices of real algebraic curves. Discrete Comput. Geom. 11, 65–81 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. Silhol, R.: Hyperbolic lego and algebraic curves in genus 2 and 3. Contemporary Math. 311. Complex Manifolds and Hyperbolic Geometry, 313–334 (2001)

    Google Scholar 

  28. Yamamoto, H..: An example of a non-classical Schottky group. Duke Math. J. 63, 193–197 (1991)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Correspondence to Rubén A. Hidalgo .

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Hidalgo, R.A., Seppälä, M. (2011). Numerical Schottky Uniformizations: Myrberg’s Opening Process. In: Bobenko, A., Klein, C. (eds) Computational Approach to Riemann Surfaces. Lecture Notes in Mathematics(), vol 2013. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17413-1_6

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