Advertisement

Perspectives

  • Sebastian Klein
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2229)

Abstract

In this book we studied (singularity-free) simply periodic solutions \(u:X \to \mathbb {C}\) of the sinh-Gordon equation via their spectral data (Σ, D) in the style of Bobenko. In particular we gained insight into the asymptotic behavior of the spectral data. As we saw in Chap.  2, real-valued such solutions give rise to minimal immersions f : X → S3, and similarly to constant mean curvature (CMC) immersions into \(\mathbb {R}^3\) and H3. In this situation, umbilical points of the immersion correspond to coordinate singularities of the solution u. In the following, I sketch how the research described in this book might be extended to solutions u with such coordinate singularities, and why the corresponding minimal immersions with umbilical points are of interest.

References

  1. [Hi]
    N.J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere. J. Differ. Geom. 31, 627–710 (1990)MathSciNetCrossRefGoogle Scholar
  2. [Ho]
    H. Hopf, Differential Geometry in the Large. Lecture Notes in Mathematics, vol. 1000 (Springer, Berlin, 1983)CrossRefGoogle Scholar
  3. [Pi-S]
    U. Pinkall, I. Sterling, On the classification of constant mean curvature tori. Ann. Math. 130, 407–451 (1989)MathSciNetCrossRefGoogle Scholar
  4. [St]
    K. Strebel, Quadratic Differentials (Springer, Berlin, 1984)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Sebastian Klein
    • 1
  1. 1.School of Business Informatics & MathematicsUniversity of MannheimMannheimGermany

Personalised recommendations