In this paper, using a doubling technique, a Riemannian Penrose-type inequality is obtained for asymptotically flat half-spaces obeying appropriate positivity properties.
More precisely, let \((M,g)\) be a Riemannian \(n\)-manifold in the usual dimension range, i.e., \(3\leq n\leq 7\), satisfying two assumptions. The first assumption is that \((M,g)\) is an asymptotically flat half-space, which means that it contains an asymptotically flat region defined in the usual way, i.e., by fixing a non-compact coordinate chart in \(M\) and declaring the fall-off properties of the metric and its curvature in it, moreover it has a non-compact boundary \(\partial M\). The second assumption is that it contains a horizon boundary, i.e., a two-sided closed minimal hypersurface \(\Sigma\subset M\) such that the scalar curvature of \(g\) is non-negative along the corresponding exterior region, i.e., the non-compact connected component of \(M\setminus\Sigma\), necessarily involving the previous asymptotically flat region, moreover the mean curvature of the boundary portion of \(M\) belonging to this exterior region is non-negative. For a more precise description of this situation see the introduction of the paper. Finally introduce the mass \(m(g)\) of \((M,g)\) by an expression which seems to be an appropriate generalization of the usual ADM mass, see Equation (2) in the paper. Then authors’ main result is that under these circumstances the inequality \[ m(g)\leq\left(\frac{1}{2}\right)^{\frac{n}{n-1}} \left(\frac{\mathrm{Area}_g(\Sigma )}{\omega_{n-1}}\right)^{\frac{n-2}{n-1}} \] holds, where \(\omega_{n-1}\) is the area of the Euclidean \((n-1)\)-sphere \(S^{n-1}\subset{\mathbb R}^n\), with equality if and only if \((M,g)\) is isometric to the Schwarzschild half-space, see Corollary 10 and Theorem 12 in the paper.
The proof is based on “doubling” \((M,g)\), i.e., gluing two copies of \((M,g)\) together along their boundaries \(\partial M\) with appropriate induced orientations and then “smoothing off” the resulting data along this gluing hypersurface.