2022/12/20 by Luca Benatti, Benatti, Luca, Mattia Fogagnolo +3 · 1 citation
Mathematics · #31C12 #53C21 #53E10 #83C99 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2212.10215
openalex publication_date 2022/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the Riemannian Penrose Inequality holds for Asymptotically Flat 3-manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the ADM mass being a well-defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential-theoretic version of it, recently introduced by Agostiniani, Oronzio and the third named author. As a consequence, we establish the equality between ADM mass and Huisken's Isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose Inequality in terms of the Isoperimetric mass on any 3-manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well-posed notion of weak Inverse Mean Curvature Flow. In particular, such Isoperimetric Riemannian Penrose Inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's Isoperimetric mass and the Hawking mass.