Rigidity of Free Boundary Minimal Disks in Mean Convex Three-Manifolds

The purpose of this article is to study rigidity of free boundary minimal two-disks that locally maximize the modified Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature and mean convex boundary. Assuming the strict stability of \(\Sigma \) $\mathrm{\Sigma }$ , we prove that a neighborhood of it in M is isometric to one of the half de Sitter–Schwarzschild space.