On a duality property of isothermic surfaces

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Isothermic parameterizations are synonyms of isothermal curvature line parameterizations, for surfaces immersed in Euclidean spaces. This short article presents a conjugation relationship between the mean curvature and the Hopf differential which correspond to a pair of dual isothermic surfaces, f and f, respectively. This relationship is natural, considering that integrable surfaces are defined as solutions of integrable systems (Lax systems based on moving frames). For any given Riemannian metric, every integrable surface is born from a couple of parents: the mean curvature H and the Hopf function Q. For any two isothermic surfaces that are dual to one another, the couple of parents is essentially the same, but the mother and father reverse roles, which leads to a conjugation formula for H, Q, H and Q, that is proven in an elementary way.

Original languageEnglish
Pages (from-to)85-90
Number of pages6
JournalJP Journal of Geometry and Topology
Issue number1
StatePublished - 2017


  • Duality
  • Hopf differential
  • Isothermic coordinates
  • Isothermic surfaces
  • Mean curvature


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