Lower semicontinuity of the ADM mass in dimensions two through seven

TL;DR

This paper extends the understanding of how the ADM mass behaves under local convergence of asymptotically flat metrics in dimensions two through seven, with implications for general relativity and scalar curvature.

Contribution

It generalizes semicontinuity results of the ADM mass to higher dimensions and under different convergence types, building on recent geometric inequalities.

Findings

Semicontinuity holds in dimensions 3 to 7 for pointed convergence with asymptotically Schwarzschild limits.

Semicontinuity also holds under weighted convergence in all dimensions ≥ 3, with a simpler proof.

In 2D, the asymptotic cone angle replaces ADM mass in semicontinuity considerations.

Abstract

The semicontinuity phenomenon of the ADM mass under pointed (i.e., local) convergence of asymptotically flat metrics is of interest because of its connections to nonnegative scalar curvature, the positive mass theorem, and Bartnik's mass-minimization problem in general relativity. In this paper, we extend a previously known semicontinuity result in dimension three for $C^2$ pointed convergence to higher dimensions, up through seven, using recent work of S. McCormick and P. Miao (which itself builds on the Riemannian Penrose inequality of H. Bray and D. Lee). For a technical reason, we restrict to the case in which the limit space is asymptotically Schwarzschild. In a separate result, we show that semicontinuity holds under weighted, rather than pointed, $C^2$ convergence, in all dimensions $n \geq 3$, with a simpler proof independent of the positive mass theorem. Finally, we also address the two-dimensional case for pointed convergence, in which the asymptotic cone angle assumes the role of the ADM mass.