Hypersurfaces with constant mean curvature in spheres

Abstract:

Let ${M^n}$ be a compact hypersurface of a sphere with constant mean curvature $H$. We introduce a tensor $\phi$, related to $H$ and to the second fundamental form, and show that if ${\left | \phi \right |^2} \leqslant {B_H}$, where ${B_H} \ne 0$ is a number depending only on $H$ and $n$, then either ${\left | \phi \right |^2} \equiv 0$ or ${\left | \phi \right |^2} \equiv {B_H}$. We also characterize all ${M^n}$ with ${\left | \phi \right |^2} \equiv {B_H}$.