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Abstract: 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.
Adams et al. (Tue,) studied this question.