Area estimates and rigidity of non-compact 𝐻-surfaces in 3-manifolds

Abstract

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 counterexample that although that area estimate holds for minimal surfaces, one does not have rigidity for equality in this case.