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.