Rigidity of marginally outer trapped surfaces in charged initial data sets

We investigate marginally outer trapped surfaces (MOTS) \(\Sigma ^2\) within a three-dimensional initial data set \(M^3\), devoid of charge density, for the Einstein–Maxwell equations in the absence of a magnetic field and with a cosmological constant \(\Lambda \). Assuming \(\Sigma \) to be a stable MOTS with genus \(g(\Sigma )\), we derive an inequality that relates the area of \(\Sigma \), \(g(\Sigma )\), \(\Lambda \), and the charge \(q(\Sigma )\) of \(\Sigma \). In cases where equality is achieved, we demonstrate local splitting of M along \(\Sigma \). Specifically, in the scenario where \(\Lambda >0\), we establish that \(\Sigma \) forms a round 2-sphere. These findings extend the theorems of Galloway and Mendes to initial data sets featuring an electric field. Moreover, for \(\Lambda >0\), we additionally demonstrate that these initial data sets can be locally embedded as spacelike hypersurfaces within the Charged Nariai spacetime.