Classification of Willmore $2$-spheres in $S^n$

Abstract

This paper resolves a long-standing open problem by providing a classification of Willmore $2$-spheres in $S^n$. We show that any such $2$-sphere is either totally isotropic--originating from the projection of a special twistor curve in the twistor bundle over an even-dimensional sphere--or strictly $k$-isotropic, obtained via $(m-k)$ steps of adjoint transforms of a strictly $m$-isotropic minimal surface in $\mathbb{R}^n$, where $0\leq k\leq m\leq[n/2]-2$. Our approach hinges on the construction of a harmonic sequence in the real Grassmannian over the Lorentz space, derived from the harmonic conformal Gauss map of the original Willmore sphere. This sequence terminates finitely and generalizes, in part, the classical theory of harmonic sequences for harmonic $2$-spheres in complex Grassmannians.