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

TL;DR

This work analyzes area-charge inequalities for charged initial data in Einstein-Maxwell spacetimes and deriving local rigidity when equality is attained. By leveraging the charged dominant energy condition and MOTS stability, it first establishes an infinitesimal rigidity result and then propagates it to a neighborhood via a foliation by constant null mean curvature surfaces, showing the geometry splits as a Riemannian product with electric and magnetic fields normal to the foliation. It treats both the time-symmetric and general initial data cases, obtaining precise constraints on the second fundamental form and matter densities in the equality scenario. An explicit Bertotti-Robinson spacetime with dyionic fields is provided to demonstrate sharpness of the rigidity results.

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 $\mathcal{A} \geq 4π(\mathcal{Q}_{\rm E}^2 + \mathcal{Q}_{\rm M}^2)$ 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.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1
• proof
• proof : Proof of Theorem \ref{['thm:main2']}
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['thm:main1']}