Let $M^n$, $n\ge3$, be a compact differentiable manifold with nonpositive Yamabe invariant $��(M)$. Suppose $g_0$ is a continuous metric with $V(M, g_0)=1$, smooth outside a compact set $��$, and is in $W^{1,p}_{loc}$ for some $p>n$. Suppose the scalar curvature of $g_0$ is at least $��(M)$ outside $��$. We prove that $g_0$ is Einstein outside $��$ if the codimension of $��$ 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 $��$ is a compact embedded hypersurface, and $g_0$ is smooth up to $��$ from two sides of $��$, and if the difference of the mean curvatures along $��$ at two sides of $��$ has a fixed appropriate sign. Then $g_0$ is also Einstein outside $��$. 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 $2$.

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