A Minkowski-type inequality in the AdS-Melvin space

TL;DR

This paper establishes a sharp Minkowski-type inequality for surfaces in the AdS-Melvin space, extending geometric inequalities to a charged, asymptotically AdS spacetime using symmetry assumptions and a weighted flow.

Contribution

It proves a Minkowski-type inequality in AdS-Melvin space for general surfaces, building on special symmetric cases and employing a novel weighted normal flow method.

Findings

Proved the inequality for axisymmetric surfaces.

Extended the inequality to small perturbations of tori.

Demonstrated the inequality for general surfaces using a weighted flow.

Abstract

The AdS-Melvin spacetime was introduced by Astorino and models the AdS soliton with electromagnetic charge. It is a static spacetime with a time-symmetric Cauchy hypersurface, which we refer to as the AdS-Melvin space. In this paper, we study a sharp Minkowski-type inequality for surfaces embedded in the AdS-Melvin space. We first prove the inequality for special cases in which the surface enjoys axisymmetry or is a small perturbation of a coordinate torus. We then use a weighted normal flow to show that the inequality holds for general surfaces.