The Penrose Inequality for Metrics with Singular Sets

TL;DR

This paper extends the understanding of the Penrose inequality by analyzing metrics with lower-dimensional singular sets, removing previous assumptions on the nature and conditions of the singularities.

Contribution

It provides a new result on the Penrose inequality for metrics with singular sets of dimension less than n-1, without extra conditions on mean curvature.

Findings

Established the Penrose inequality for metrics with lower-dimensional singular sets.

Extended previous results by removing assumptions on the singular set's hypersurface nature.

Contributed to the rigidity case analysis for such metrics.

Abstract

We study the Penrose inequality and its rigidity for metrics with singular sets. Our result could be viewed as a complement of Theorem 1.1 of Lu and Miao (J. Funct. Anal. 281, 2021) and Theorem 1.2 of Shi, Wang and Yu (Math. Z. 291, 2019), in which they assume the singular set is a hypersurface and assume an additional condition on the mean curvature. As a complement, this paper study the case of singular set of dimensional less than n-1, without any additional conditions.