Mathematische Zeitschrift

, Volume 245, Issue 1, pp 63–91 | Cite as

Hyperbolic constant mean curvature one surfaces: Spinor representation and trinoids in hypergeometric functions

  • Alexander I. Bobenko
  • Tatyana V. Pavlyukevich
  • Boris A. Springborn


Hypergeometric Function Spinor Representation 


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  1. 1.
    Biswas, I.: A criterion for the existence of a parabolic stable bundle of rank two over the projective line. Int. J. Math. 9, 523–533 (1998)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bobenko, A.I.: Surfaces in terms of 2 by 2 matrices. Old and new integrable cases. In: A. Fordy, J. Wood, eds. Harmonic maps and integrable systems, Vieweg, Brauschweig, 1994, pp. 83–127Google Scholar
  3. 3.
    Bryant, R.L.: Surfaces of mean curvature one in hyperbolic space. Astérisque, 154–155, 321–347 (1987)Google Scholar
  4. 4.
    Collin, P., Hauswirth, L., Rosenberg, H.: The geometry of finite topology Bryant surfaces. Ann. Math. (2) 153, 623–659 (2001)Google Scholar
  5. 5.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions. Vol. I, McGraw-Hill Book Co., New York, 1953Google Scholar
  6. 6.
    Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori. Invent. Math. 146, 507–503 (2001)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Klein, F.: Vorlesungen über die hypergeometrische Funktion, Springer-Verlag, Berlin, 1981. Reprint of the 1933 originalGoogle Scholar
  8. 8.
    Kusner, R., Schmitt, N.: The spinor representation of surfaces in space, arXiv:dg-ga/9610005, 1996Google Scholar
  9. 9.
    Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and theorems for the special functions of mathematical physics, Springer-Verlag, Berlin, 1966Google Scholar
  10. 10.
    Rossman, W., Umehara, M., Yamada, K.: Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus. Tôhoku Math. J. 49, 449–484 (1997)MATHGoogle Scholar
  11. 11.
    Umehara, M., Yamada, K.: Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space. Ann. Math. 137, 611–638 (1993)MathSciNetMATHGoogle Scholar
  12. 12.
    Umehara, M., Yamada, K.: Surfaces of constant mean curvature c in H 3(-c 2) with prescribed hyperbolic Gauss map. Math. Ann. 304, 203–209 (1996)MathSciNetMATHGoogle Scholar
  13. 13.
    Umehara, M., Yamada, K.: Another construction of a CMC-1 surface in H 3. Kyungpook Math. J. 35, 831–849 (1996)MathSciNetMATHGoogle Scholar
  14. 14.
    Umehara, M., Yamada, K.: Metrics of constant curvature 1 with three conical singularities on the 2-sphere. Illinois J. Math. 44, 72–94 (2000)MathSciNetMATHGoogle Scholar
  15. 15.
    Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge University Press, Cambridge, 1962Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alexander I. Bobenko
    • 1
  • Tatyana V. Pavlyukevich
    • 1
  • Boris A. Springborn
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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