Area-charge inequality and local rigidity in charged initial data sets

TL;DR

This paper explores the geometric implications of equality in area-charge inequalities for certain surfaces in Einstein-Maxwell initial data, establishing rigidity results that characterize the local geometry when the inequality is saturated.

Contribution

It proves new rigidity theorems linking equality in area-charge bounds to specific geometric structures in Einstein-Maxwell initial data sets.

Findings

Saturation of the inequality implies the electric and magnetic fields are normal to the surface.

The local geometry near the surface is isometric to a Riemannian product.

Rigidity results apply both in time-symmetric and non-time-symmetric cases.

Abstract

This paper investigates the geometric consequences of equality in area-charge inequalities for spherical minimal surfaces and, more generally, for marginally outer trapped surfaces (MOTS), within the framework of the Einstein-Maxwell equations. We show that, under appropriate energy and curvature conditions, saturation of the inequality $\mathcal{A} \geq 4\pi(\mathcal{Q}_{\rm E}^2 + \mathcal{Q}_{\rm M}^2)$ imposes a rigid geometric structure in a neighborhood of the surface. In particular, the electric and magnetic fields must be normal to the foliation, and the local geometry is isometric to a Riemannian product. We establish two main rigidity theorems: one in the time-symmetric case and another for initial data sets that are not necessarily time-symmetric. In both cases, equality in the area-charge bound leads to a precise characterization of the intrinsic and extrinsic geometry of the initial data near the critical surface.