The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass

TL;DR

This paper proves a Penrose inequality for asymptotically locally hyperbolic spaces with negative mass using inverse mean curvature flow, extending understanding of geometric inequalities in these settings.

Contribution

It introduces a Penrose inequality applicable to negative mass metrics in asymptotically locally hyperbolic spaces, a novel result in this geometric context.

Findings

Proves a Penrose inequality for negative mass metrics

Uses inverse mean curvature flow as the main technique

Links the inequality to static uniqueness of negative mass Kottler metrics

Abstract

In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature at least -6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes from a previous result of P. Chru\'sciel and W. Simon, which states that the Penrose inequality we prove implies a static uniqueness theorem for negative mass Kottler metrics.