Hypersurfaces with constant higher order mean curvature in space forms

Let Mcn+1, n≥3, be a space form of constant sectional curvature c=0,1,−1 and M a complete oriented hypersurface of Mcn+1 having constant r-th mean curvature Hr for some 2≤r≤n−1 and two principal curvatures of multiplicities (n−1) and 1. We suppose further that |Hr|>0 for c=0, |Hr|>1 for c=−1 and being Hr any value for c=1. We prove that the infimum and the supremum of the squared norm of the second fundamental form of M are attained, obtain sharp bounds for them and characterize those hypersurfaces where the bounds is attained.