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 $\ddot{I}{ƒ}_k=\frac{{\int }_M(|A{|}^2{)}^k}{|M|}$. We show that $\ddot{I}{ƒ}_k\ge 2^k$ provided $n=2$ and $M$ is not totally geodesic. When $n=4$ and $M$ has two distinct principal curvatures, we show $\ddot{I}{ƒ}_2\ge 16$. When $n\ge 3$ and $M$ has two distinct principal curvatures, for each integer $k\ge 2$, there exists a positive constant $\hat{I}{´}_k(n)<n$, if $|A{|}^2\ge \hat{I}{´}_k(n)$, we have $\ddot{I}{ƒ}_k\ge n^k$. All the equality holds iff $M$ is isometric to a Clifford hypersurface.

7 pages, no figure