# Lower semicontinuity of ADM mass under intrinsic flat convergence

## TL;DR

This paper proves that the ADM mass, a key quantity in general relativity, is lower semicontinuous under the weak intrinsic flat convergence of asymptotically flat manifolds, extending previous results to a broader convergence setting.

## Contribution

It establishes lower semicontinuity of ADM mass under intrinsic flat convergence, using Huisken's isoperimetric mass as a substitute, and introduces new convergence results for subregions of integral current spaces.

## Key findings

- Lower semicontinuity of ADM mass under intrinsic flat convergence.
- Introduction of asymptotically flat local integral current spaces.
- Convergence results for subregions of $ ext{F}$-converging integral current spaces.

## 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.