Area Estimates and Rigidity of Non-compact $H$-Surfaces in 3-Manifolds

For appropriately values of $H$, we obtain an area estimate for a complete non-compact $H$-surface of finite topology and finite area, embedded in a three-manifold of negative curvature. Moreover, in the case of equality and under additional assumptions, we prove that a neighbourhood of the mean convex side of the surface must be isometric to a hyperbolic Fuchsian manifold. Also, we show by an counter-example that although that area estimate holds for minimal surfaces, one does not have rigidity for equality in this case.