Classification of Willmore 2-spheres in the 5-dimensional sphere

2014/09/08 by Ma, Xiang, Wang, Changping, Wang, Peng · 1 citation
#53A10 #53C42 #53C45 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1409.2427

Abstract

The classification of Willmore 2-spheres in the n-dimensional sphere Sⁿ 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) super-conformal surfaces in S⁴; (2) minimal surfaces in R⁵; (3) adjoint transforms of super-conformal minimal surfaces in R⁵. 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 2-spheres in S⁵ of type (3), we describe all adjoint transforms of a super-conformal minimal surfaces in Rⁿ and provide some explicit criterions on the immersion property. As an application, we obtain new immersed Willmore 2-spheres in S⁵ and S⁶, which are not S-Willmore.

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• Classification of Willmore 2-spheres in Sⁿ

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