Willmore 2-spheres in \(S^n\): a survey. (English) Zbl 1350.53017
Futaki, Akito (ed.) et al., Geometry and topology of manifolds. 10th China-Japan geometry conference for friendship, held in Shanghai, China, September 7–11, 2014. Tokyo: Springer (ISBN 978-4-431-56019-7/hbk; 978-4-431-56021-0/ebook). Springer Proceedings in Mathematics & Statistics 154, 211-233 (2016).
Summary: We give an overview of the classification problem of Willmore 2-spheres in \(S^n\), and report the recent progress on this problem when \(n=5\) (or even higher). We explain two main ingredients in our work. The first is the adjoint transform of Willmore surfaces introduced by the first author, which generalizes the dual Willmore surface construction. The second is the DPW method applied to Willmore surfaces whose conformal Gauss map is well-known to be a harmonic map into a non-compact symmetric space (a joint work of J. F. Dorfmeister and the second author [“Harmonic maps of finite uniton type into non-compact inner symmetric spaces”, Preprint, arXiv:1305.2514]). We also sketch a possible way to classify all Willmore 2-spheres in \(S^n\).
For the entire collection see [Zbl 1350.58001].
For the entire collection see [Zbl 1350.58001].
MSC:
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 58E20 | Harmonic maps, etc. |
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
References:
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