  Motivated by the rigidity case in the localized Riemannian Penrose
inequality, we show that suitable singular metrics attaining the optimal value
in the Riemannian Penrose inequality is necessarily smooth in properly
specified coordinates. If applied to hypersurfaces enclosing the horizon in a
spatial Schwarzschild manifold, the result gives the rigidity of isometric
hypersurfaces with the same mean curvature.
