Asymptotically flat half-spaces (M,g) are asymptotically flat manifolds with a non-compact boundary. They naturally arise as suitable subsets of initial data for the Einstein Field equations.
In this talk, I will present a proof of the Riemannian Penrose inequality for asymptotically flat half-spaces with horizon boundary (joint with M. Eichmair) that works in all dimensions up to seven. This inequality gives a sharp bound for the area of the horizon boundary in terms of the half-space mass of (M,g). To prove the inequality, we double (M, g) along its non-compact boundary and smooth the doubled manifold appropriately. To prove rigidity, we use variational methods to show that, if equality holds, the non-compact boundary of (M,g) must be totally geodesic.
I will also explain how our techniques can be used to prove rigidity for the Riemannian Penrose inequality for asymptotically flat manifolds.
