A Volumetric Penrose Inequality for Conformally Flat Manifolds

Annales Henri Poincaré - Tập 12 - Trang 67-76 - 2010
Fernando Schwartz1
1Department of Mathematics, University of Tennessee, Knoxville, USA

Tóm tắt

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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Annales Henri Poincaré

Tập 12

67-76

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