Removable singularity of positive mass theorem with continuous metrics

TL;DR

This paper proves the positive mass theorem for asymptotically flat manifolds with continuous metrics that are smooth away from a singular set, extending previous results and confirming a conjecture for the case of continuous metrics.

Contribution

It extends the positive mass theorem to non-spin manifolds with continuous metrics and minimal singular sets, confirming a conjecture for the $p= olinebreak\infty$ case.

Findings

Nonnegative ADM mass for manifolds with singularities.

Rigidity result: zero mass implies Euclidean space.

Extension of positive mass theorem to non-spin, low-regularity metrics.

Abstract

In this paper, we consider asymptotically flat Riemannnian manifolds $(M^n,g)$ with $C^0$ metric $g$ and $g$ is smooth away from a closed bounded subset $\Sigma$ and the scalar curvature $R_g\ge 0$ on $M\setminus \Sigma$. For given $n\le p\le \infty$, if $g\in C^0\cap W^{1,p}$ and the Hausdorff measure $\mathcal{H}^{n-\frac{p}{p-1}}(\Sigma)<\infty$ when $n\le p<\infty$ or $\mathcal{H}^{n-1}(\Sigma)=0$ when $p=\infty$, then we prove that the ADM mass of each end is nonnegative. Furthermore, if the ADM mass of some end is zero, then we prove that $(M^n,g)$ is isometric to the Euclidean space by showing the manifold has nonnegative Ricci curvature in RCD sense. This extends the result of [Lee-LeFloch2015] from spin to non-spin, also improves the result of [Shi-Tam2018] and [Lee2013]. Moreover, for $p=\infty$, this confirms a conjecture of Lee [Lee2013].