  Let ${\mathscr F}(N\times \mathbb{R})$ be the set of all closed
$H$-hypersurfaces $M\subset N\times \mathbb{R}$, where $N$ is a simply
connected complete Riemannian $n$-manifold with sectional curvature $K_{N}\leq
-\kappa^{2}<0$. We show that ${\Hm}(N\times \mathbb{R})=\inf_{M\in {\mathscr
F}(N\times \mathbb{R})}\{| H_{M}| \}\geq (n-1)\kappa/n $.
