A new characterization for Clifford hypersurfaces

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 ${\sigma }_k={\displaystyle \frac{{\int }_M(\mathrm{\mid }A{\mathrm{\mid }}^2{)}^k}{\mathrm{\mid }M\mathrm{\mid }}}$. 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 $\mathrm{\mid }A{\mathrm{\mid }}^2\ge {\delta }_k(n)$, we have ${\sigma }_k\ge n^k$. All the equality holds iff $M$ is isometric to a Clifford hypersurface.