Embedded CMC Hypersurfaces on Hyperbolic Spaces

in this paper we will prove that for every integer n>1, there exists a real number h0<-1 such that every h∈ (-∞,h0) can be realized as the mean curvature of an embedding of hn-1\times s1 in the n+1-dimensional space hn+1. for n=2 we explicitly compute the value h0. for a general value n, we provide a function ξn defined on (-∞,-1), which is easy to compute numerically, such that, if ξn(h)>-2π, then, h can be realized as the mean curvature of an embedding of hn-1\times s1 in the (n+1)-dimensional space hn+1.