Penrose-type inequalities with a Euclidean background

TL;DR

This paper introduces a new Penrose-like inequality for conformally flat manifolds valid in all dimensions, providing sharper bounds on ADM mass and extending to metrics with scalar-flat backgrounds and zero area singularities.

Contribution

It presents a novel Penrose-type inequality applicable in all dimensions for conformally flat manifolds, improving upon the classical RPI by using Euclidean areas and generalizing to scalar-flat backgrounds.

Findings

The inequality is sharper than RPI when many minimal surfaces are present.

It applies to manifolds of any dimension, not limited to up to seven.

A new lower bound for ADM mass with zero area singularities is established.

Abstract

The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background, and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.