Willmore Hypersurfaces with vanishing Mobius form in 𝕊4

Let [Formula: see text] be an immersed hypersurface without umbilical points, one can define the Möbius metric [Formula: see text] and the Möbius form [Formula: see text], which are invariant under the Möbius transformation group of [Formula: see text]. A hypersurface is called a Willmore hypersurface if it is the critical point of the volume functional of [Formula: see text] with respect to the Möbius metric [Formula: see text]. In this paper, we classify completely the Willmore hypersurfaces with vanishing Möbius form (i.e., [Formula: see text]) in [Formula: see text] up to a Möbius transformation of [Formula: see text].