A Green's function proof of the Positive Mass Theorem

Abstract

In this short note, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green's function of an asymptotically flat \(3\)-manifolds. In the same context and for \(1<p<3\), a Geroch-type calculation is performed along the level sets of \(p\)-harmonic functions, leading to a new proof of the Riemannian Penrose Inequality in some case studies.