# Perspectives

## 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* → *S*^{3}, and similarly to constant mean curvature (CMC) immersions into \(\mathbb {R}^3\) and *H*^{3}. 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

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