New energy inequality in AdS spacetimes

We study time symmetric initial data for asymptotically anti--de Sitter spacetimes with conformal boundary containing a spatial circle. Such $d$-dimensional initial datasets can contain ($d\ensuremath{-}2$)-dimensional minimal surfaces if the circle is contractible. We compute the minimum energy of a large class of such initial data as a function of the area $A$ of this minimal surface. The statement $E\ensuremath{\ge}{E}_{\mathrm{min}}(A)$ is analogous to the Penrose inequality which bounds the energy from below by a function of the area of a ($d\ensuremath{-}1$)-dimensional minimal surface.