The Penrose inequality is a conjecture stating that the total mass of an asymptotically flat spacetime is at least as large as the mass of the black holes it contains. In the Riemannian setting — that is, for time-symmetric spacetimes — this inequality has been proven through various methods. A notable approach by Huisken and Ilmanen (2001) relies on the monotonicity of Hawking mass along inverse mean curvature flow. More recently, Agostiniani, Mantegazza, Mazzieri, and Oronzio (2022) provided a proof using monotonicity formulas in the framework of nonlinear potential theory. However, both techniques require stronger assumptions on the metric than those needed to state the problem. In this talk, I will present a unified perspective on the two approaches. This synergy allows to extend the applicability of this inequality to more general settings.

This talk is based on a series of joint papers with  with M. Fogagnolo (UNIPD), L. Mazzieri (UNITN), A. Pluda (UNIPI) and M. Pozzetta (UNIMI).