Lower semicontinuity of ADM mass under intrinsic flat convergence

Abstract

A natural question in mathematical general relativity is how the ADM mass behaves as a functional on the space of asymptotically flat 3-manifolds of nonnegative scalar curvature. In previous results, lower semicontinuity has been established by the first-named author for pointed \(C^2\) convergence, and more generally by both authors for pointed \(C^0\) convergence (all in the Cheeger-Gromov sense). In this paper, we show this behavior persists for the much weaker notion of pointed Sormani–Wenger intrinsic flat (\({\mathcal {F}}\)) volume convergence, under natural hypotheses. We consider smooth manifolds converging to asymptotically flat local integral current spaces (a new definition), using Huisken’s isoperimetric mass as a replacement for the ADM mass. Along the way we prove results of independent interest about convergence of subregions of \({\mathcal {F}}\)-converging sequences of integral current spaces.