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 a embedding of H^{n-1}\times S^1 in the (n+1)-dimensional spaces H^{n+1}. For $n=2$ we explicitly compute the value H_0. For a general value n, we provide 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 a embedding of H^{n-1}\times S^1 in the (n+1)-dimensional spaces H^{n+1}.