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

TL;DR

This paper establishes new inequalities linking boundary capacity and quasi-local mass in asymptotically flat 3-manifolds, introduces singular fill-ins with potential independent interest, and provides novel characterizations of Schwarzschild manifolds.

Contribution

It introduces new inequalities relating boundary capacity and quasi-local mass, and explores singular fill-ins with nonnegative scalar curvature, offering fresh insights into Schwarzschild manifolds.

Findings

Derived inequalities connecting boundary capacity and quasi-local mass.

Introduced singular fill-ins with nonnegative scalar curvature.

Provided new variational characterizations of Schwarzschild manifolds.

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.