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Discrete special isothermic surfaces

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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.

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    F. Burstall

  2. Vienna University of Technology, Wiedner Hauptstraße 8–10/104, 1040, Vienna, Austria

    U. Hertrich-Jeromin

  3. Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501, Japan

    W. Rossman

  4. Departamento de Matemática, CMAF, Universidade de Lisboa, 1749-016, Lisbon, Portugal

    S. Santos

Authors
  1. F. Burstall
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  2. U. Hertrich-Jeromin
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  3. W. Rossman
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  4. S. Santos
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Correspondence to U. Hertrich-Jeromin.

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Burstall, F., Hertrich-Jeromin, U., Rossman, W. et al. Discrete special isothermic surfaces. Geom Dedicata 174, 1–11 (2015). https://doi.org/10.1007/s10711-014-0001-4

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  • DOI: https://doi.org/10.1007/s10711-014-0001-4

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