A Green’s Function Proof of the Positive Mass Theorem

Abstract

In this paper, 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 $$1<p<3$$ 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 under favourable assumptions. A new characterisation of scalar curvature lower bounds in terms of the monotonicity formulas is also given.