Addendum to “Complete rotation hypersurfaces with H _k constant in space forms”*

Abstract.

Here we study complete rotation hypersurfaces with constant k-th mean curvature H _k in $\mathbb{S}^{{n + 1}},k$ even and 2 < k < n. We prove the existence of a constant $H^{0}_{k} < 0$ such that there are no such hypersurfaces for $H_{k} < H^{0}_{k}$ . We have only one compact hypersurface of this kind with $H_{k} = H^{0}_{k}$ . For each $H^{0}_{k} < H_{k} < 0$ there is a corresponding family of complete immersed rotation hypersurfaces, each family containing two isoparametric hypersurfaces. For H _k ≥ 0, there is also such a family, now containing only one isoparametric hypersurface. Finally, we prove the existence of compact hypersurfaces with arbitrarily large H _k , neither isometric to a sphere nor to a product of spheres.