Riemannian Penrose inequality via Nonlinear Potential Theory

TL;DR

This paper introduces a novel proof of the Riemannian Penrose inequality using nonlinear potential theory, specifically through a new monotonicity formula related to the $p$-capacitary potential in asymptotically flat 3-manifolds.

Contribution

It presents a new proof technique for the Riemannian Penrose inequality based on a monotonicity formula involving the $p$-capacitary potential, expanding the mathematical tools available for this problem.

Findings

Established a new monotonicity formula for $p$-capacitary potential.

Proved the Riemannian Penrose inequality for a class of initial data.

Extended the applicability of potential theory methods in geometric inequalities.

Abstract

We provide a new proof of the Riemannian Penrose inequality for time-symmetric asymptotically flat initial data with a single black-hole horizon. The proof proceeds through a newly established monotonicity formula holding along the level sets of the $p$-capacitary potential of the horizon boundary, in any asymptotically flat $3$-manifold with nonnegative scalar curvature.