Near-equality of the Penrose Inequality for rotationally symmetric Riemannian manifolds

TL;DR

This paper investigates the near-equality cases of the Penrose Inequality for rotationally symmetric, asymptotically flat Riemannian manifolds, showing they are close to Schwarzschild spaces with appended cylinders in Lipschitz and Intrinsic Flat distances.

Contribution

It establishes that near-equality of the Penrose Inequality implies the manifold is close to a Schwarzschild space with a cylinder, extending previous results on the Positive Mass Theorem.

Findings

Manifolds close to equality resemble Schwarzschild spaces with cylinders

Lipschitz Distance bounds Intrinsic Flat Distance in this context

Near-equality implies geometric closeness to model spaces

Abstract

This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the Lipschitz Distance bounds the Intrinsic Flat Distance on compact sets, we also obtain a result for Intrinsic Flat Distance, which is a more appropriate distance for more general near-equality results, as discussed in [LS]