A Möbius rigidity of compact Willmore hypersurfaces in Sⁿ⁺¹

Let x:Mⁿ→Sⁿ⁺¹ be an immersed hypersurface without umbilical point, then one can define the Möbius metric g, the Möbius second fundamental form B and the Blaschke tensor A on the hypersurface Mⁿ 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 prove that if the hypersurface x is a compact Willmore hypersurface without umbilical point, then [Formula presented] the equality holds if and only if the hypersurface Mⁿ is Möbius equivalent to one of the Willmore tori [Formula presented] where the tensor [Formula presented].