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

Abstract

For appropriate values of H H , we obtain an area estimate for a complete non-compact H H -surface of finite topology and finite area, embedded in a 3-manifold of negative curvature. Moreover, in the case of equality and under additional assumptions, we prove that a neighborhood of the mean convex side of the surface must be isometric to a hyperbolic Fuchsian manifold. Also, we provide a counterexample showing that, in the case of minimal surfaces, equality in the area estimate does not necessarily imply a local rigidity result for the ambient manifold.