Rigidity of Riemannian Penrose inequality with corners and its implications

We study suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality. More precisely, we demonstrate that the singular metric is necessarily smooth in properly specified coordinates. When applied to hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result gives the rigidity of isometric hypersurfaces with the same mean curvature.