Rigidity of minimal hypersurfaces of spheres with two principal curvatures (Q1881524)

Let \(M\) be a compact minimal hypersurface of the sphere \({\mathbb S}^n, n>3\) with two different principal curvatures everywhere. It is shown that \(\int_M \| A \| ^2 \leq (n-1)| M | \) where \(\| A \| ^2\) is the square of the norm of the shape operator \(A\) and\(| M | \) denotes the volume of \(M\) and that the equality holds only when \(M\) is a Clifford hypersurface.