A $\sigma_{2}$ Penrose inequality for conformal asymptotically hyperbolic 4-discs

TL;DR

This paper establishes a Penrose inequality for conformal metrics on 4-discs with asymptotically hyperbolic ends, linking mass and singularities under a specific curvature bound, and classifies the equality case.

Contribution

It introduces a new Penrose inequality for conformal asymptotically hyperbolic 4-discs with singularities, under a curvature condition, and characterizes the equality case as hyperbolic space.

Findings

Proved a Penrose type inequality relating mass and singularities.

Classified the equality case as hyperbolic 4-space.

Curvature condition implies non-positive energy density.

Abstract

In this paper, we consider conformal metrics on a unit 4-disc with an asymptotically hyperbolic end and possible isolated conic singularities. We define a mass term of the AH end. If the $\sigma_{2}$ curvature has lower bound $\sigma_{2}\geq\frac{3}{2}$, we prove a Penrose type inequality relating the mass and contributions from singularities. We also classify sharp cases, which is the standard hyperbolic 4-space $\mathbb{H}^{4}$ when no singularity occurs. It is worth noting that our curvature condition implies non-positive energy density.