Modified Hawking mass and rigidity of three-manifolds with boundary

In this paper, we prove a rigidity result for three-dimensional Riemannian manifolds with boundary, under the assumption that a free boundary minimal two-disk, which locally maximizes a modified Hawking mass, is embedded in a $3$-dimensional Riemannian manifold with negative scalar curvature and mean convex boundary. First, we establish area estimates for free boundary strictly stable two-disks. Finally, we show that the $3$-dimensional Riemannian manifold with boundary is locally isometric to the half anti-de Sitter-Schwarzschild manifold.