Minimal Darboux transformations

  • Udo Hertrich-Jeromin
  • Atsufumi HondaEmail author
Original Paper


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.


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 



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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Copyright information

© The Managing Editors 2016

Authors and Affiliations

  1. 1.Vienna University of TechnologyViennaAustria
  2. 2.Miyakonojo CollegeNational Institute of TechnologyMiyakonojoJapan

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