The Penrose inequality on perturbations of the Schwarzschild exterior

We prove a version the Penrose inequality for black hole space-times which are perturbations of the Schwarzschild exterior in a slab around a null hypersurface $\underline{\mathcal{N}}_0$. $\underline{\mathcal{N}}_0$ terminates at past null infinity $\mathcal{I}^-$ and $\mathcal{S}_0:=\partial\underline{\mathcal{N}}_0$ is chosen to be a marginally outer trapped sphere. We show that the area of $\mathcal{S}_0$ yields a lower bound for the Bondi energy of sections of past null infinity, thus also for the total ADM energy. Our argument is perturbative, and rests on suitably deforming the initial null hypersurface $\underline{\mathcal{N}}_0$ to one for which the natural "luminosity" foliation originally introduced by Hawking yields a monotonically increasing Hawking mass, and for which the leaves of this foliation become asymptotically round. It is to ensure the latter (essential) property that we perform the deformation of the initial nullhypersurface $\underline{\mathcal{N}}_0$.