A new energy inequality in AdS

We study time symmetric initial data for asymptotically AdS spacetimes with conformal boundary containing a spatial circle. Such $d$-dimensional initial data sets can contain $(d-2)$-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 $E\ge E_{min}(A)$ is analogous to the Penrose inequality which bounds the energy from below by a function of the area of a $(d-1)$-dimensional minimal surface.