A remark on compact hypersurfaces with constant mean curvature in space forms

Abstract

In this note we characterize compact hypersurfaces of dimension $n\ge 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when $n\ge 3$ and $H\ne 0$, they are locally contained in a rotational hypersurface. In dimension two, the integral pinching condition reduces to a topological assumption and we recover the classical Hopf-Chern result.