Rigidity of Free Boundary Surfaces via a Modified Charged Hawking Mass

Abstract

We introduce a modified charged Hawking mass for free boundary surfaces in time-symmetric three-dimensional Einstein–Maxwell initial data sets with a cosmological constant, and we investigate the rigidity of free boundary minimal two-disks that locally maximize this mass in Riemannian three-manifolds for which R-2|E|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R-2|E|^2$$\end{document} is bounded below by a positive constant and whose boundary is mean-convex. Assuming that Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} is strictly stable, we prove that a neighborhood of Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} in M is isometric to a half de Sitter Reissner–Nordström space. As an application, we establish a sharp area-charge inequality and characterize the equality cases, thereby obtaining rigidity results for half charged Nariai spaces and related geometries.