Minimal Darboux transformations

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.

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References

  1. Bianchi, L.: Il teorema di permutabilità per le trasformazioni di Darboux delle superficie isoterme. Rom. Acc. L. Rend. 13, 359–367 (1904)

    MATH  Google Scholar 

  2. Burstall, F.E., Hertrich-Jeromin, U., Pedit, F., Pinkall, U.: Curved flats and isothermic surfaces. Math. Z. 225, 199–209 (1997)

    MathSciNet  Article  MATH  Google 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)

  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)

  5. Christoffel, E.: Ueber einige allgemeine Eigenschaften der Minimumsflächen. J. Reine Angew. Math. 67, 218–228 (1867)

    MathSciNet  Article  Google Scholar 

  6. Darboux, G.: Sur les surfaces isothermiques. C.R. Acad. Sci. Paris 128, 1299–1305, 1538 (1899)

  7. Ferus, D., Pedit, F.: Curved flats in symmetric spaces. Manuscr. Math. 91, 445–454 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  8. Goursat, E.: Sur un mode de transformation des surfaces minima. Acta Math. 11, 135–186, 257–264 (1887)

  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)

    MathSciNet  MATH  Google Scholar 

  10. Hertrich-Jeromin, U., Pedit, F.: Remarks on the Darboux transform of isothermic surfaces. Doc. Math. 2, 313–333 (1997)

    MathSciNet  MATH  Google Scholar 

  11. Hertrich-Jeromin, U.: Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  12. Kokubu, M., Umehara, M., Yamada, K.: Flat fronts in hyperbolic 3-space. Pacif. J. Math. 216, 149–175 (2004)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  14. Nitsche, J.C.C.: Lectures on minimal surfaces, vol. 1. Cambridge University Press, Cambridge (1989)

    Google Scholar 

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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.

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Correspondence to Atsufumi Honda.

Additional information

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). https://doi.org/10.1007/s13366-016-0301-y

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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