Removable singularity of positive mass theorem with continuous metrics (Q2172494)

We say that \((M^n,g)\), an \(n\)-dimensional smooth manifold endowed with a \(C^0\)-metric \(g=(g_{ij})_{1\le i,j\le n}\), is asymptotically flat if there is a compact subset \(K\subset M^n\) such that \(g\) is \(C^2\) on \(M^n\setminus K\), which has a finite family of components \((\Sigma_l)_{1\le l\le p}\), called ends, and there is a \((\mathbb R^n\setminus B)\)-valued diffeomorphism map \(\psi_l\) on \(\Sigma_l\) where \(B\) is a Euclidean ball. When \(\psi_l\) is seen as coordinate system on each end \(\Sigma_l\), we have \(g_{ij}-\delta_{ij}=O(|x|^{-\rho})\), \(g_{ij,k}=O(|x|^{-\rho-1})\), and \(g_{ij,kl}=O(|x|^{-\rho-2})\) where \(\rho\) is a constant bigger than \(\frac{n-2}{2}\). The Arnowitt-Deser-Misner mass (ADM mass) of each \(\Sigma_l\) is defined as the limit of \[\frac{\omega_{n-1}}{2(n-1)}\int_{S_r}\sum_{i,j=1}^n(g_{ij,i}-g_{ii,j})\nu^jd\mu,\text{ as }r \text{ goes to infinity},\] where \(\nu\) is the unit outward normal vector of \(S_r\), the sphere of radius \(r\) in \(\Sigma_l\), such that \(\omega_{n-1}\) is it the surface area. Let \(R_g\) be the scalar curvature of \(g\). The authors state that if \(g\in C^0\cap W^{1,p}(M^n)\) for \(p\in [n,\infty]\) of an symptotically flat manifold \((M^n,g)\) and smooth away from a bounded closed subset \(\Sigma\) with either a finite \((n-\frac{p}{p-1})\)-dimensional Hausdorff measure for \(p<\infty\) or a null \((n-1)\)-dimensional Hausdorff measure such that \(R_g\) is positive on \(M^n\setminus\Sigma\), then the ADM mass of \(g\) of each end is positive. Furthermore, the ADM mass of one end is zero if and only if \((M^n, g)\) is isometric to a Euclidean space.