Minimal Darboux transformations


We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats in the 2-sphere and flat fronts in hyperbolic space.

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The second author expresses his gratitude to the members of the Institute of Discrete Mathematics and Geometry at TU Wien for their hospitality during his stay in 2015/16.

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Correspondence to Atsufumi Honda.

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This work has been partially supported by the Austrian Science Fund (FWF) and the Japan Society for the Promotion of Science (JSPS) through the FWF/JSPS Joint Project Grant I1671-N26 “Transformations and Singularities”.

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Hertrich-Jeromin, U., Honda, A. Minimal Darboux transformations. Beitr Algebra Geom 58, 81–91 (2017).

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  • Minimal surface
  • Darboux transformation
  • Christoffel transformation
  • Goursat transformation
  • Bianchi permutability
  • Riccati equation
  • Flat front
  • Curved flat
  • Hyperbolic geometry

Mathematics Subject Classification

  • Primary 53A10
  • 37K35
  • Secondary 53C42
  • 53A30
  • 37K25
  • 34M45