Hypersurfaces with constant scalar curvature in space forms

In this paper we study the rigidity of complete hypersurfaces with constant scalar curvature in Riemannian space forms. Under an appropriate constraint on Φ, the traceless part of its second fundamental form, we prove that either the hypersurface is totally umbilical or it holds a sharp estimate for the supremum of the norm of Φ, with equality if and only if the hypersurface is isoparametric with two distinct principal curvatures. Moreover, we also construct complete non-isoparametric rotational examples which show that our constraint on Φ is sharp and necessary.