ADM mass and the capacity-volume deficit at infinity

Abstract

Based on the isoperimetric inequality, G. Huisken proposed a definition of total mass in general relativity that is equivalent to the ADM mass for (smooth) asymptotically flat 3-manifolds of nonnegative scalar curvature, but that is well-defined in greater generality. In a similar vein, we use the isocapacitary inequality (bounding capacity from below in terms of volume) to suggest a new definition of total mass. We prove an inequality between it and the ADM mass, and prove the reverse inequality with harmonically flat asymptotics, or, with general asymptotics, for exhaustions by balls (as opposed to arbitrary compact sets). This approach to mass may have applications to problems involving low regularity metrics and convergence in general relativity, and may have some advantages relative to the isoperimetric mass. Some conjectures, analogs of known results for CMC surfaces and isoperimetric regions, are proposed.