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