The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary

TL;DR

This paper proves the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary using a new approximation scheme for free boundary inverse mean curvature flow, extending previous work and supporting conjectures in geometric analysis.

Contribution

It introduces a novel approximation method for free boundary inverse mean curvature flow and establishes the monotonicity of a free boundary Hawking mass, advancing the understanding of Penrose inequalities.

Findings

Proved the Riemannian Penrose inequality for manifolds with non-compact boundary.

Developed a new approximation scheme for free boundary inverse mean curvature flow.

Established monotonicity of a free boundary Hawking mass.

Abstract

In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation scheme for the weak free boundary inverse mean curvature flow, introduced by Marquardt in 2012, and establish the monotonicity of a free boundary version of the Hawking mass. Our result also implies a non-optimal Penrose inequality for asymptotically flat support surfaces in $\mathbb{R}^3$ and thus sheds some light on a conjecture made by Huisken.