Abstract
We consider asymptotically flat Riemannian manifolds with non-negative scalar curvature that are conformal to \({\mathbb{R}^{n}{\setminus} \Omega, n\ge 3}\), and so that their boundary is a minimal hypersurface. (Here, \({\Omega\subset \mathbb{R}^{n}}\) is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by \({\frac{1}{2}\left(V/\beta_{n}\right)^{(n-2)/n}}\), where V is the Euclidean volume of Ω and β n is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga (Commun. Anal. Geom. 10:999–1016, 2002). Surprisingly, we do not require the boundary to be outermost.
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Communicated by Piotr T. Chrusciel
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Schwartz, F. A Volumetric Penrose Inequality for Conformally Flat Manifolds. Ann. Henri Poincaré 12, 67–76 (2011). https://doi.org/10.1007/s00023-010-0070-3
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DOI: https://doi.org/10.1007/s00023-010-0070-3