Embedded CMC Hypersurfaces on Hyperbolic Spaces

In this paper we will prove that for every integer $n>1$, there exists a real number $H_0<-1$ such that every $H\in (-\infty,H_0)$ can be realized as the mean curvature of an embedding of $H^{n-1}\times S^1$ in the $n+1$-dimensional space $H^{n+1}$. For $n=2$ we explicitly compute the value $H_0$. For a general value $n$, we provide a function $\xi_n$ defined on $(-\infty,-1)$, which is easy to compute numerically, such that, if $\xi_n(H)>-2\pi$, then, $H$ can be realized as the mean curvature of an embedding of $H^{n-1}\times S^1$ in the $(n+1)$-dimensional space $H^{n+1}$.