A ${\sigma }_2$ Penrose inequality for conformal asymptotically hyperbolic 4-discs

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 ${\sigma }_2\ge \frac{3}{2}$, 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.

References in corpus (1)

• Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

Cited by in corpus (4)

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• On the ${\sigma }_k$-Nirenberg problem
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• A Note On Andrews Inequality