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June 2017 Classification of Willmore two-spheres in the 5-dimensional sphere
Xiang Ma, Changping Wang, Peng Wang
J. Differential Geom. 106(2): 245-281 (June 2017). DOI: 10.4310/jdg/1497405626


The classification of Willmore two-spheres in the $n$-dimensional sphere $S^n$ is a long-standing problem, solved only when $n = 3, 4$ by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when $n = 5$. There are three types of such surfaces up to Möbius transformations: (1) superconformal surfaces in $S^4$; (2) minimal surfaces in $R^5$; (3) adjoint transforms of superconformal minimal surfaces in $R^5$. In particular, Willmore surfaces in the third class are not $S$-Willmore (i.e., without a dual Willmore surface).

To show the existence of Willmore two-spheres in $S^5$ of type (3), we describe all adjoint transforms of a superconformal minimal surface in $R^n$ and provide some explicit criterions on the immersion property. As an application, we obtain new immersed Willmore two-spheres in $S^5$ and $S^6$, which are not $S$-Willmore.


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Xiang Ma. Changping Wang. Peng Wang. "Classification of Willmore two-spheres in the 5-dimensional sphere." J. Differential Geom. 106 (2) 245 - 281, June 2017.


Received: 26 February 2015; Published: June 2017
First available in Project Euclid: 14 June 2017

zbMATH: 06846951
MathSciNet: MR3662992
Digital Object Identifier: 10.4310/jdg/1497405626

Rights: Copyright © 2017 Lehigh University


Vol.106 • No. 2 • June 2017
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