A local rigidity theorem for minimal two-spheres in an electrovacuum spacetime

The purpose of this article is to prove that, under suitable constrains on the electrovacuum spacetime $M$, if $Σ\subset M$ is an embedded strictly stable minimal two-sphere which locally maximizes the charged Hawking mass, then there exist a neighborhood of it in $M$ isometric to the Reissner-Nordström-de Sitter space. At the same time, motived by \cite{Brendle}, we will deduce an estimate for area of a two-sphere which is locally area minimizing in an electrovacuum spacetime. Moreover, if the equality holds, then there exist a neighborhood of it in $M$ isometric to the charged Nariai space.

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