Abstract: In this talk we describe a new monotonicity formula holding along the level sets of the Green's function of an asymptotically flat 3-manifold with nonnegative scalar curvature. Using such a formula, we obtain a simple proof of the celebrated positive mass theorem. 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.
The results are obtained in collaboration with V. Agostiniani and F. Oronzio.