On compact CMC-Hypersurfaces of $N\times \mathbb{R}$

Abstract

Let $\mathcal{F}(N\times \mathbb{R})$ 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 $K_N\le -{\kappa }^2<0$. We show that ${\Hm}(N\times \mathbb{R})=\inf_{M\in {\mathscr F}(N\times \mathbb{R})}\{| H_{M}| \}\geq (n-1)κ/n $.

We changed the title from a Remark on compact CMC-Hypersurfaces of $N\times \mathbb{R}$ to On compact and we added another theorem to the paper (theorem 1.4)