preprint

Inserted: 21 dec 2022

Year: 2022

Abstract:

We show that the Riemannian Penrose Inequality holds on Asymptotically Flat $3$-manifolds with nonnegative scalar curvature and connected horizon boundary, under the minimal decay assumptions that are needed for the $\mathrm{ADM}$ mass to be well defined. The proof exploits new asymptotic comparison arguments involving the notion of Isoperimetric mass, originally introduced by Huisken. As a byproduct, we obtain a Riemannian Penrose Inequality in terms of the Isoperimetric mass, holding on any Asymptotically Flat $3$-manifold with nonnegative scalar curvature and connected horizon boundary, on which a well posed notion of weak Inverse Mean Curvature Flow is available. Similar results are provided in the case of $\mathscr{C}^0$-Asymptotically Schwarzschildian $3$-manifolds.