Geometriae Dedicata

, Volume 174, Issue 1, pp 1–11 | Cite as

Discrete special isothermic surfaces

  • F. Burstall
  • U. Hertrich-Jeromin
  • W. Rossman
  • S. Santos
Original Paper

Abstract

We discuss special isothermic nets of type \(N\), a new class of discrete isothermic nets, generalizing isothermic nets with constant mean curvature in spaceforms. In the case \(N=2\) these are the discrete analogues of Bianchi’s special isothermic surfaces that can be regarded as the origin of the rich transformation theory of isothermic surfaces. Accordingly, special isothermic nets come with Bäcklund transformations and a Lawson correspondence. The notion of complementary nets naturally occurs and sheds further light on the relation between geometry and integrability.

Keywords

Isothermic surface Darboux transformation Lawson correspondence Bäcklund transformation Polynomial conserved quantity Constant mean curvature 

Mathematics Subject Classification

53A10 53C42 53A30 37K25 37K35 

Notes

Acknowledgments

The third author expresses his gratitude to Vienna University of Technology for financial support and their hospitality during the preparation of this paper. The figures in this text were created using Mathematica.

References

  1. 1.
    Bernstein, H.: Non-special, non-canal isothermic tori with spherical lines of curvature. Trans. AMS 353, 2245–2274 (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Bianchi, L.: Ricerche sulle superficie isoterme e sulla deformazione delle quadriche. Ann. Math. 11, 93–157 (1904)MATHGoogle Scholar
  3. 3.
    Bobenko, A., Pinkall, U.: Discretization of surfaces and integrable systems. Oxf. Lect. Ser. Math. Appl. 16, 3–58 (1999)MathSciNetGoogle Scholar
  4. 4.
    Bobenko, A., Suris, Y.: Discrete Differential Geometry. Integrable structure; Grad Stud Math 98. Amer Math Soc, Providence, RI (2008)Google Scholar
  5. 5.
    Bobenko, A., Hertrich-Jeromin, U., Lukyanenko, I.: Discrete constant mean curvature nets in space forms: Steiner’s formula and Christoffel duality. Discret. Comput. Geom. (2014)Google Scholar
  6. 6.
    Burstall, F., Hertrich-Jeromin, U., Rossman, W., Santos, S.: Discrete surfaces of constant mean curvature. RIMS Kokyuroku 1880, 133–179 (2014)Google Scholar
  7. 7.
    Burstall, F., Santos, S.: Special isothermic surfaces of type \(d\). J. Lond. Math. Soc. 85, 571–591 (2012)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Burstall, F., Hertrich-Jeromin, U., Rossman, W.: Discrete linear Weingarten surfaces. arXiv:1406.1293 (2014)
  9. 9.
    Eisenhart, L.: Transformations of Surfaces. Princeton Univ Press, Princeton (1923)MATHGoogle Scholar
  10. 10.
    Hertrich-Jeromin, U., Hoffmann, T., Pinkall, U.: A discrete version of the Darboux transform for isothermic surfaces. Oxf. Lect. Ser. Math. Appl. 16, 59–81 (1999)MathSciNetGoogle Scholar
  11. 11.
    Hertrich-Jeromin, U.: Introduction to Möbius Differential Geometry; London Math Soc Lect Note Series 300. Cambridge Univ Press, Cambridge (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • F. Burstall
    • 1
  • U. Hertrich-Jeromin
    • 2
  • W. Rossman
    • 3
  • S. Santos
    • 4
  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Vienna University of TechnologyWiedner Hauptstraße 8–10/104ViennaAustria
  3. 3.Department of MathematicsKobe UniversityKobeJapan
  4. 4.Departamento de Matemática, CMAFUniversidade de LisboaLisbonPortugal

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