Willmore surfaces in spheres via loop groups II: a coarse classification of Willmore two-spheres by potentials

Applying the DPW version of the theory developed by Burstall and Guest for harmonic maps of finite uniton type, we derive a coarse classification of Willmore two-spheres in $S^{n+2}$ in terms of the normalized potential of their (harmonic) conformal Gauss maps. Moreover, for the case of $S^6$, some geometric properties of the corresponding Willmore two-spheres are discussed. The classical classification of Willmore two-spheres in $S^4$ are also derived as a corollary.