Totally umbilic surfaces in hyperbolic 3-manifolds of finite volume

For every connected surface S of finite negative Euler characteristic and every H∈ [0,1), we construct a hyperbolic 3-manifold $N(S,H)$ of finite volume and a proper, two-sided, totally umbilic embedding f S→ N(S,H) with mean curvature H. Conversely, we prove that a complete, totally umbilic surface with mean curvature H ∈ [0,1) embedded in a hyperbolic 3-manifold of finite volume must be proper and have finite negative Euler characteristic.