Removable singularity of positive mass theorem with continuous metrics

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 $\mathrm{\Sigma }$ and the scalar curvature $R_g\ge 0$ on $M\setminus \mathrm{\Sigma }$. For given $n\le p\le \mathrm{\infty }$, if $g\in C^0\cap W^{1,p}$ and the Hausdorff measure ${\mathcal{H}}^{n-\frac{p}{p-1}}(\mathrm{\Sigma })<\mathrm{\infty }$ when $n\le p<\mathrm{\infty }$ or ${\mathcal{H}}^{n-1}(\mathrm{\Sigma })=0$ when $p=\mathrm{\infty }$, 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 $p=\mathrm{\infty }$, this confirms a conjecture of Lee [Lee2013].