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24 June 2024
 
  » arxiv » 2302.00175

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Doubling of asymptotically flat half-spaces and the Riemannian Penrose inequality
Michael Eichmair ; Thomas Koerber ;
Date 1 Feb 2023
AbstractBuilding on previous works of H. L. Bray, of P. Miao, and of S. Almaraz, E. Barbosa, and L. L. de Lima, we develop a doubling procedure for asymptotically flat half-spaces $(M,g)$ with horizon boundary $Sigmasubset M$ and mass $minmathbb{R}$. If $3leq dim(M)leq 7$, $(M,g)$ has non-negative scalar curvature, and the boundary $partial M$ is mean-convex, we obtain the Riemannian Penrose-type inequality $$ mgeqleft(frac{1}{2} ight)^{frac{n}{n-1}},left(frac{|Sigma|}{omega_{n-1}} ight)^{frac{n-2}{n-1}} $$ as a corollary. Moreover, in the case where $partial M$ is not totally geodesic, we show how to construct local perturbations of $(M,g)$ that increase the scalar curvature. As a consequence, we show that equality holds in the above inequality if and only if the exterior region of $(M,g)$ is isometric to a Schwarzschild half-space. Previously, these results were only known in the case where $dim(M)=3$ and $Sigma$ is a connected free boundary hypersurface.
Source arXiv, 2302.00175
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