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

Abstract

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.