This work contains essentially two rigidity results.
For compact surfaces with boundary properly embedded in a Riemannian three-manifold with mean convex boundary that are local minima for the free boundary problem for the area, we prove that a geometric invariant constructed from the infimum of the scalar curvature of the ambient manifold, the infimum of the mean curvature of its boundary, the area of the surface and the length of its boundary is bounded from above by a constant which depends only on the topology of the surface, with equality (under additional hypotheses) if and only if the surface has constant Gaussian curvature, its boundary has constant geodesic curvature and the metric of the ambient three-manifold locally splits as a product metric in a neighborhood of the surface.
For asymptotically hyperbolic three-manifolds that have a minimal boundary, scalar curvature greater than or equal to -6 and that are sufficiently small perturbations of the Anti-de Sitter-Schwarzschild spaces of positive mass, we prove that the Hawking mass of the boundary (which is a function of the area of the boundary only)
is bounded from above by the mass of the manifold (which depends only on the geometry at infinity), with equality if and only if the manifold is isometric to the Anti-de Sitter-Schwarzschild space of same mass. This proves the Penrose Conjecture for this class of asymptotically hyperbolic manifolds.