Rigidity of Free Boundary Surfaces via a Modified Charged Hawking Mass

We introduce a modified charged Hawking mass for free boundary surfaces in time-symmetric three-dimensional Einstein–Maxwell initial data sets with a cosmological constant, and we investigate the rigidity of free boundary minimal two-disks that locally maximize this mass in Riemannian three-manifolds for which R-2|E|² R - 2 | E | 2 is bounded below by a positive constant and whose boundary is mean-convex. Assuming that Σ Σ is strictly stable, we prove that a neighborhood of Σ Σ in M is isometric to a half de Sitter Reissner–Nordström space. As an application, we establish a sharp area-charge inequality and characterize the equality cases, thereby obtaining rigidity results for half charged Nariai spaces and related geometries.