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

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 Mobius 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.