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We construct for every connected surface $S$ of finite negative Eulercharacteristic and every $H \in [0,1)$, a hyperbolic 3-manifold $N(S,H)$ offinite volume and a proper, two-sided, totally umbilic embedding $f\colon S\toN(S,H)$ with mean curvature $H$. Conversely, we prove that a complete, totallyumbilic surface with mean curvature $H \in [0,1)$ embedded in a hyperbolic3-manifold of finite volume must be proper and have finite negative Eulercharacteristic.

Totally umbilic surfaces in hyperbolic 3-manifolds of finite volume