Riemannian Penrose inequality in all dimensions

Abstract

We prove the Riemannian Penrose inequality in arbitrary dimension for smooth complete asymptotically flat manifolds with nonnegative scalar curvature and compact outer-minimizing minimal boundary, where the boundary is allowed to have a singular set of Hausdorff dimension at most \(n-8\). Moreover, the equality holds exactly when the manifold is isometric to the Riemannian Schwarzschild exteriors. Our proof extends Bray's conformal-flow method to higher dimensions, where the outer-minimizing enclosures along the flow may be singular.