A new characterization for Clifford hypersurfaces

Abstract

For a closed minimal immersed hypersurface $M$ in $\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\ge 2$, define a constant $\sigma_k=\dfrac{\int_M (|A|^2)^k}{|M|}$. We show that $\sigma_k \ge 2^k$ provided $n=2$ and $M$ is not totally geodesic. When $n=4$ and $M$ has two distinct principal curvatures, we show $\sigma_2 \ge 16$. When $n\ge 3$ and $M$ has two distinct principal curvatures, for each integer $k\ge 2$, there exists a positive constant $\delta_k(n)<n$, if $|A|^2\ge \delta_k(n)$, we have $\sigma_k\ge n^k$. All the equality holds iff $M$ is isometric to a Clifford hypersurface.