A positive mass theorem for low-regularity Riemannian metrics

TL;DR

This paper extends the positive mass theorem to low-regularity Riemannian metrics in Sobolev spaces, showing it holds under weaker smoothness assumptions for certain manifolds and providing approximation results.

Contribution

It proves the positive mass theorem for continuous metrics in $W^{2, n/2}_{loc}$ and establishes approximation by smooth metrics with non-negative scalar curvature.

Findings

Positive mass theorem holds for low-regularity metrics in $W^{2, n/2}_{loc}$.

Metrics with non-negative scalar curvature can be approximated by smooth metrics.

Provides lower bounds on ADM mass for metrics with scalar curvature failing to be non-negative.

Abstract

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ for manifolds of dimension less than or equal to $7$ or spin-manifolds of any dimension. More generally, we give a (negative) lower bound on the ADM mass of metrics for which the scalar curvature fails to be non-negative, where the negative part has compact support and sufficiently small $L^{n/2}$ norm. We show that a Riemannian metric in $W^{2, p}_{loc}$ for some $p > \frac{n}{2}$ with non-negative scalar curvature in the distributional sense can be approximated locally uniformly by smooth metrics with non-negative scalar curvature. For continuous metrics in $W^{2, n/2}_{loc}$, there exist smooth approximating metrics with non-negative scalar curvature that converge in $L^p_{loc}$ for all $p < \infty$.