Compact Willmore hypersurfaces with two distinct principal curvatures in Sn+1Sn+1

Let x:Mn→Sn+1x:Mn→Sn+1 be a compact Willmore hypersurface with two distinct principal curvatures. In this paper, we present a classification of the compact Willmore hypersurfaces, which multiplicities of principal curvatures are greater than one. And if the one of principal curvatures is simple, we give an integral inequality involving the Möbius scalar curvature of x  . Particularly, if the Möbius scalar curvature SgSg is constant, then Sg=n−1k(n−k)−3n−2n2, and x(Mn)x(Mn) is Möbius equivalent to Sk(n−kn)×Sn−k(kn), 1⩽k