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

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.