Scalar curvature and singular metrics

Let $M^n$, $n\ge 3$, be a compact differentiable manifold with nonpositive Yamabe invariant $\sigma (M)$. Suppose $g_0$ is a continuous metric with $V(M, g_0)=1$,smoothoutsideacompactset$\Sigma$,andisin$W^{1,p}_{loc}$ for some $p>n$. Suppose the scalar curvature of $g_0$ is at least $\sigma (M)$ outside $\mathrm{\Sigma }$. We prove that $g_0$ is Einstein outside $\mathrm{\Sigma }$ if the codimension of $\mathrm{\Sigma }$ is at least $2$. If in addition, $g_0$ is Lipschitz then $g_0$ is smooth and Einstein after a change the smooth structure. If $\mathrm{\Sigma }$ is a compact embedded hypersurface, and $g_0$ is smooth up to $\mathrm{\Sigma }$ from two sides of $\mathrm{\Sigma }$, and if the difference of the mean curvatures along $\mathrm{\Sigma }$ at two sides of $\mathrm{\Sigma }$ has a fixed appropriate sign. Then $g_0$ is also Einstein outside $\mathrm{\Sigma }$. For manifolds with dimension between $3$ and $7$ without spin assumption, we obtain a positive mass theorem on an asymptotically flat manifold for metrics with a compact singular set of codimension at least $2$.