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Minimal Darboux transformations

  • Udo Hertrich-Jeromin
  • Atsufumi HondaEmail author
Original Paper
  • 96 Downloads

Abstract

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.

Keywords

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 

Notes

Acknowledgments

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.

References

  1. Bianchi, L.: Il teorema di permutabilità per le trasformazioni di Darboux delle superficie isoterme. Rom. Acc. L. Rend. 13, 359–367 (1904)zbMATHGoogle Scholar
  2. Burstall, F.E., Hertrich-Jeromin, U., Pedit, F., Pinkall, U.: Curved flats and isothermic surfaces. Math. Z. 225, 199–209 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Burstall, F.E.: Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems, Integrable systems, geometry, and topology, AMS/IP Stud. Adv. Math., vol. 36, pp. 1–82. Am. Math. Soc., Providence, RI (2006)Google Scholar
  4. Burstall, F.E., Hertrich-Jeromin U., Rossman, W.: Lie geometry of flat fronts in hyperbolic space. C.R. Acad. Sci. Paris 348, 661–664 (2010)Google Scholar
  5. Christoffel, E.: Ueber einige allgemeine Eigenschaften der Minimumsflächen. J. Reine Angew. Math. 67, 218–228 (1867)MathSciNetCrossRefGoogle Scholar
  6. Darboux, G.: Sur les surfaces isothermiques. C.R. Acad. Sci. Paris 128, 1299–1305, 1538 (1899)Google Scholar
  7. Ferus, D., Pedit, F.: Curved flats in symmetric spaces. Manuscr. Math. 91, 445–454 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Goursat, E.: Sur un mode de transformation des surfaces minima. Acta Math. 11, 135–186, 257–264 (1887)Google Scholar
  9. Hertrich-Jeromin, U.: Supplement on curved flats in the space of point pairs and isothermic surfaces: a quaternionic calculus. Doc. Math. 2, 335–350 (1997)MathSciNetzbMATHGoogle Scholar
  10. Hertrich-Jeromin, U., Pedit, F.: Remarks on the Darboux transform of isothermic surfaces. Doc. Math. 2, 313–333 (1997)MathSciNetzbMATHGoogle Scholar
  11. Hertrich-Jeromin, U.: Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  12. Kokubu, M., Umehara, M., Yamada, K.: Flat fronts in hyperbolic 3-space. Pacif. J. Math. 216, 149–175 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Martínez, A., Roitman, P., Tenenblat, K.: A connection between flat fronts in hyperbolic space and minimal surfaces in euclidean space. Ann. Glob. Anal. Geom. 48, 233–254 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Nitsche, J.C.C.: Lectures on minimal surfaces, vol. 1. Cambridge University Press, Cambridge (1989)Google Scholar

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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