A volumetric Penrose inequality for conformally flat manifolds

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We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to ⁿ Ω, n≥ 3, and so that their boundary is a minimal hypersurface. (Here, Ω⊂ ⁿ is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by (V/βₙ)⁽ⁿ⁻²⁾/ⁿ, where V is the Euclidean volume of Ω and βₙ is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga brayiga. Surprisingly, we do not require the boundary to be outermost.