ADM mass, area and capacity in asymptotically flat $3$-manifolds with nonnegative scalar curvature

TL;DR

This paper improves existing inequalities relating mass, capacity, and area in asymptotically flat 3-manifolds with nonnegative scalar curvature, using monotonicity formulas along harmonic potential level sets.

Contribution

It extends Bray's sharp mass-capacity inequality and Bray-Miao's capacity-area bounds under new topological and curvature assumptions.

Findings

Enhanced mass-capacity inequality for asymptotically flat manifolds.

Improved capacity-area bounds for boundary surfaces.

Utilization of monotonicity formulas along harmonic potential level sets.

Abstract

We show an improvement of Bray sharp mass-capacity inequality and Bray-Miao sharp upper bound of the capacity of the boundary in terms of its area, for three-dimensional, complete, one-ended asymptotically flat manifolds with compact, connected boundary and with nonnegative scalar curvature, under appropriate assumptions on the topology and on the mean curvature of the boundary. Our arguments relies on two monotonicity formulas holding along level sets of a suitable harmonic potential, associated to the boundary of the manifold. This work is an expansion of the results contained in the PhD thesis of the author.