Implications of some mass-capacity inequalities (Q6564120)

A smooth Riemannian \(3\)-manifold \((M, g)\) is called asymptotically flat (AF) if \(M\), outside a compact set, is diffeomorphic to \(\mathbb{R}^{3}\) minus a ball; the metric coefficients satisfy\N\[\Ng_{ij} = \delta_{ij} + O(|x|^{-\tau}), \partial g_{ij} = O(|x|^{-\tau-1}), \partial^{2} g_{ij} = O(|x|^{-\tau-2}),\N\]\Nfor some \(\tau > 1/2\); and the scalar curvature of g is integrable. Under these conditions, the limit\N\[\Nm=\lim_{r\to +\infty} \frac{1}{16\pi}\int_{|x|=r} \sum_{j,k} (g_{jk,j}-g_{jj,k})\frac{x^k}{|x|}\N\]\Nexists and is called the ADM mass of \((M, g)\). A fundamental result on the mass and the scalar curvature is the Riemannian positive mass theorem, asserting that for a complete, asymptotically flat \(3\)-manifold with nonnegative scalar curvature without boundary, we have \(m\geq 0\) and equality holds if and only if the manifold is isometric to \(\mathbb{R}^{3}\). \N\NThe author finds sufficient conditions that imply the nonnegativity as well as positive lower bounds of the ADM mass, on a class of manifolds with nonnegative scalar curvature, with or without a singularities.