Mass, capacitary functions, and the mass-to-capacity ratio

TL;DR

This paper explores the relationship between ADM mass, harmonic functions, and capacity in asymptotically flat 3-manifolds, deriving new formulas, inequalities, and conditions that enhance understanding of mass positivity and geometric properties.

Contribution

It introduces new formulas linking ADM mass with harmonic functions, establishes monotone quantities and inequalities, and provides novel conditions for mass positivity in asymptotically flat manifolds.

Findings

New formulas for ADM mass via harmonic functions

Monotone quantities and geometric inequalities for manifolds with simple topology

Mass-to-capacity ratio bounded below by a function of boundary geometry

Abstract

We study connections among the ADM mass, positive harmonic functions tending to zero at infinity, and the capacity of the boundary of asymptotically flat $3$-manifolds with nonnegative scalar curvature. First we give new formulae that detect the ADM mass via harmonic functions. Then we derive a family of monotone quantities and geometric inequalities if the underlying manifold has simple topology. As an immediate application, we observe several additional proofs of the $3$-dimensional Riemannian positive mass theorem. One proof leads to new, sufficient conditions that imply positivity of the mass via $C^0$-geometry of regions separating the boundary and $\infty$. A special case of such sufficient conditions shows, if a region enclosing the boundary has relative small volume, then the mass is positive. As further applications, we obtain integral identities for the mass-to-capacity ratio. We also promote the inequalities to become equality on spatial Schwarzschild manifolds outside rotationally symmetric spheres. Among other things, we show the mass-to-capacity ratio is always bounded below by one minus the square root of the normalized Willmore functional of the boundary. Prompted by our findings, we carry out a study of manifolds satisfying a constraint on the mass-to-capacity ratio. We point out such manifolds satisfy improved inequalities, their mass has an upper bound depending only on the boundary data, there are no closed minimal surfaces enclosing the boundary, and these manifolds include static extensions in the context of the Bartnik quasi-local mass.