On the lower semicontinuity of the ADM mass

TL;DR

This paper investigates the lower semicontinuity of the ADM mass in various geometric convergence settings, establishing conditions under which it holds and linking it to the positive mass theorem.

Contribution

It proves lower semicontinuity of the ADM mass under pointed Cheeger--Gromov convergence and rotational symmetry assumptions, connecting geometric analysis with mass stability.

Findings

Lower semicontinuity holds for n=3 under pointed Cheeger--Gromov convergence.

Under rotational symmetry, lower semicontinuity holds for weak convergence of embeddings.

The positive mass theorem can be derived from the lower semicontinuity of the ADM mass.

Abstract

The ADM mass, viewed as a functional on the space of asymptotically flat Riemannian metrics of nonnegative scalar curvature, fails to be continuous for many natural topologies. In this paper we prove that lower semicontinuity holds in natural settings: first, for pointed Cheeger--Gromov convergence (without any symmetry assumptions) for $n=3$, and second, assuming rotational symmetry, for weak convergence of the associated canonical embeddings into Euclidean space, for $n \geq 3$. We also apply recent results of LeFloch and Sormani to deal with the rotationally symmetric case, with respect to a pointed type of intrinsic flat convergence. We provide several examples, one of which demonstrates that the positive mass theorem is implied by a statement of the lower semicontinuity of the ADM mass.