Mass, Capacitary Functions, and the Mass-to-Capacity Ratio

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