Willmore hypersurfaces with vanishing Möbius form in S⁴

Let x: Mⁿ → Sⁿ⁺¹ be an immersed hypersurface without umbilical points, one can define the Möbius metric g and the Möbius form C, which are invariant under the Möbius transformation group of Sⁿ⁺¹. A hypersurface is called a Willmore hypersurface if it is the critical point of the volume functional of Mⁿ with respect to the Möbius metric g. In this paper, we classify completely the Willmore hypersurfaces with vanishing Möbius form (i.e.