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We obtain a bound for the area of a capillary H-surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side product, we obtain existence of totally geodesic embedded surfaces in hyperbolic three-manifolds under the assumption of the existence of a H-surface realizing the area bound, in particular, under the existence of a totally umbilic H-surface.

Area estimates and rigidity of capillary H-surfaces in three-manifolds with boundary