A Möbius rigidity of compact Willmore hypersurfaces in [formula omitted

Abstract

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∫Mn(‖A˜+B‖2−n−1n)dMg≥0, the equality holds if and only if the hypersurface Mn is Möbius equivalent to one of the Willmore toriWn,m=Sm(n−mn)×Sn−m(mn)↪Sn+1,1≤m≤n−1, where the tensor A˜=A+B2−1n[tr(A)+n−1n]g.