Willmore hypersurfaces with constant Möbius curvature in Rⁿ⁺¹

For an immersed hypersurface f: Mⁿ → Rⁿ⁺¹ without umbilical points, one can define the Möbius metric g on f which is invariant under the Möbius transformation group. The volume functional of g is a generalization of the well-known Willmore functional, whose critical points are called Willmore hypersurfaces. In this paper, we prove that if a n-dimensional Willmore hypersurfaces (n ≥ 3) has constant sectional curvature c with respect to g, then c = 0, n = 3, and this Willmore hypersurface is Möbius equivalent to the cone over the Clifford torus in S³ ⊂ R⁴. Moreover, we extend our previous classification of hypersurfaces with constant Möbius curvature of dimension n ≥ 4 to n = 3, showing that they are cones over the homogeneous torus S¹(r) × S¹(√1 - r²) ⊂ S³, or cylinders, cones, rotational hypersurfaces over certain spirals in the space form R ², S ², H ², respectively.