Séminaire de Géométrie
Some rigidity results for charged Riemannian manifoldsAbraao Mendes (Université Fédérale d'Alagoas)
Friday 05 December 2025 13:30 - Tours - 1180 (Bât. E2)
Résumé :
In this lecture, we explore the geometric consequences of equality in Gibbons' area-charge inequality for stable minimal 2-spheres $\Sigma^2$ in the context of the Einstein--Maxwell equations. We show that, under suitable energy (curvature) assumptions, saturation of the inequality $$ \mathcal{A}(\Sigma) \ge 4\pi \mathcal{Q}_{\rm E}^2 $$ forces a rigid geometric structure in a neighborhood of the surface $\Sigma^2$. In particular, the electric field $E$ must be normal to the foliation, and the local geometry becomes isometric to a Riemannian product.
We then extend these rigidity phenomena to both compact and non-compact time-symmetric initial data sets with boundary, establishing sharp area-charge inequalities and examining the resulting rigidity of the boundary and ambient geometries under appropriate hypotheses, including cases with non-spherical boundary topology. As time permits, we will conclude with several model examples illustrating these results.
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