Riemannian Penrose Inequality for Manifolds with Corners via Non-Linear Potential Theory

We present a new proof of the Positive Mass Theorem and the Riemannian Penrose Inequality for three-dimensional asymptotically flat Riemannian manifolds whose metrics fail to be $C^1$ across a hypersurface $\mathrm{\Sigma }$, first proven by Miao and McCormick-Miao, respectively. Unlike these approaches, ours recovers these results directly, without relying on their original formulations for smooth metrics. The proofs are based on a unified argument which applies to both theorems. We achieve this by establishing an approximate monotonicity for the quantity introduced by Agostiniani-Mantegazza-Mazzieri-Oronzio, employing the approximation scheme of Miao, for metrics with $C^{2,\alpha }$ regularity up to $\mathrm{\Sigma }$.