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

Abstract

The classification of Willmore 2-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) super-conformal surfaces in $S^4$; (2) minimal surfaces in $R^5$; (3) adjoint transforms of super-conformal 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 2-spheres in $S^5$ of type (3), we describe all adjoint transforms of a super-conformal minimal surfaces in $R^n$ and provide some explicit criterions on the immersion property. As an application, we obtain new immersed Willmore 2-spheres in $S^5$ and $S^6$, which are not S-Willmore.

28 pages. We corrected some mistakes and add the full discussions on the new examples. Comments are welcome!

References in corpus (2)

• Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms
• A class of superconformal surfaces

Cited by in corpus (2)

• Willmore surfaces in spheres via loop groups IV: on totally isotropic Willmore two-spheres in $S^6$
• On Willmore Legendrian surfaces in ${\mathbb{S}}^5$ and the contact stationary Legendrian Willmore surfaces