Riemannian Penrose inequality in all dimensions

TL;DR

This paper proves the Riemannian Penrose inequality for all dimensions in a broad class of manifolds, extending Bray's conformal-flow method to higher dimensions with singularities.

Contribution

It generalizes the Riemannian Penrose inequality to arbitrary dimensions, allowing singular boundary sets and extending Bray's method.

Findings

The inequality holds in all dimensions for the specified manifolds.

Equality occurs only for Riemannian Schwarzschild exteriors.

The proof accommodates singular outer-minimizing boundaries.

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.