Capacity, quasi-local mass, and singular fill-ins

Abstract

We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown--York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.

arXiv comment46 pages. Latest pre-print version. Final and most up-to-date version will appear in Journal fur die reine und angewandte Mathematik (Crelle)

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• Metrics with ${\lambda }_1(-\Delta +kR)\ge 0$ and flexibility in the Riemannian Penrose Inequality
• First boundary Dirac eigenvalue and boundary capacity potential