A new characterization for Clifford hypersurfaces

For a closed minimal immersed hypersurface M in Sⁿ⁺¹ with second fundamental form A, and each integer k≥ 2, define a constant σₖ=∫M (|A|²)ᵏ/|M|. We show that σₖ ≥ 2ᵏ provided $n=2$ and M is not totally geodesic. When $n=4$ and M has two distinct principal curvatures, we show σ₂ ≥ 16. When n≥ 3 and M has two distinct principal curvatures, for each integer k≥ 2, there exists a positive constant δₖ(n)<n, if |A|²≥ δₖ(n), we have σₖ≥ nᵏ. All the equality holds iff M is isometric to a Clifford hypersurface.