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

In this note we characterize compact hypersurfaces of dimension n≥2n≥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≥3n≥3 and H≠0H≠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.