The purpose of this article is study rigidity of free boundary minimal
two-disk that locally maximize the modified Hawking mass on a Riemannian
three-manifold with positive lower bound on its scalar curvature and mean
convex boundary. Under strict stability of it assumptions, we prove that a
neighborhood of it in M is isometric to one of the half de Sitter-Schwarzschild
metrics.Comment: 16 pages; no figure