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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 E₌₈₍ (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.
Horowitz et al. (Tue,) studied this question.