As on arXiv
Abstract
We prove that the Riemannian Penrose Inequality holds for Asymptotically Flat -manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the 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 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 -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.
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature
Topics Geometric inequalities, Inverse mean curvature flow, Physics · Same author Mattia Fogagnolo · Shared terms “nonnegative scalar”, “isoperimetric”, “weak inverse”
-
The asymptotic behaviour of -capacitary potentials in Asymptotically Conical manifolds
Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Same authors Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri · Shared terms “weak inverse”
-
The Riemannian Penrose Inequality with Matter Density
Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Riemannian Penrose”, “Penrose inequality”, “Huisken”
-
The Penrose inequality with charge for 2-convex initial data sets
Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Penrose inequality”, “Riemannian Penrose”, “Hawking mass”
-
Scalar curvature bounds for 3D continuous metrics through the Inverse Mean Curvature Flow
Topics Inverse mean curvature flow, Physics, Scalar curvature · Same author Mattia Fogagnolo · Shared terms “Hawking mass”, “nonnegative scalar”
-
Monotone Quantities for -Harmonic functions and the Sharp -Penrose inequality
Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Penrose inequality”, “weak inverse”, “asymptotically flat”
-
Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas
Topics Inverse mean curvature flow, Physics, Scalar curvature · Same author Luca Benatti · Shared terms “Hawking mass”, “nonnegative scalar”, “potential theoretic”
-
The -inverse mean curvature flow and the generalized Penrose conjecture
Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “Penrose”, “asymptotically flat”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6C.4The Yamabe Problem and Critical Exponents
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
- 9B.6Scalar Curvature and Topology
Positive scalar curvature, its obstructions, and the positive mass theorem.
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.