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Doubling of asymptotically flat halfspaces and the Riemannian Penrose inequality  Michael Eichmair
; Thomas Koerber
;  Date: 
1 Feb 2023  Abstract:  Building 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 halfspaces $(M,g)$ with horizon boundary $Sigmasubset M$ and mass
$minmathbb{R}$. If $3leq dim(M)leq 7$, $(M,g)$ has nonnegative scalar
curvature, and the boundary $partial M$ is meanconvex, we obtain the
Riemannian Penrosetype inequality $$
mgeqleft(frac{1}{2}
ight)^{frac{n}{n1}},left(frac{Sigma}{omega_{n1}}
ight)^{frac{n2}{n1}}
$$ 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 halfspace. 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  Services:  Forum  Review  PDF  Favorites 


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