A local rigidity theorem for minimal two-spheres in charged time-symmetric initial data set

The purpose of this article is to prove that, under suitable constraints on time-symmetric initial data set for the Einstein–Maxwell equation M, if \(\Sigma \subset M\) is an embedded strictly stable minimal two-sphere which locally maximizes the charged Hawking mass, then there exists a neighborhood of it in M isometric to the Reissner–Nordström–de Sitter space. At the same time, motivated (Bray et al. in Commun Anal Geom 18(4):821–830, 2010), we will deduce an estimate for the area of a two-sphere which is locally area minimizing on time-symmetric initial data set for the Einstein–Maxwell equation. Moreover, if the equality holds, then there exists a neighborhood of it in M isometric to the charged Nariai space.