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.
