Publication date:
April 2017
Source:Differential Geometry and its Applications, Volume 51
Author(s): Josué Meléndez, Oscar Palmas
Let ${\mathbb{M}}_c^{n+1}$, $n\ge 3$, be a space form of constant sectional curvature $c=0,1,-1$ and M a complete oriented hypersurface of ${\mathbb{M}}_c^{n+1}$ having constant r-th mean curvature $H_r$ for some $2\le r\le n-1$ and two principal curvatures of multiplicities $(n-1)$ and 1. We suppose further that $|H_r|>0$ for $c=0$, $|H_r|>1$ for $c=-1$ and being $H_r$ 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.