#### Vol. 293, No. 2, 2018

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Scalar curvature and singular metrics

### Yuguang Shi and Luen-Fai Tam

Vol. 293 (2018), No. 2, 427–470
DOI: 10.2140/pjm.2018.293.427
##### Abstract

Let ${M}^{n}$, $n\ge 3$, be a compact differentiable manifold with nonpositive Yamabe invariant $\sigma \left(M\right)$. Suppose ${g}_{0}$ is a continuous metric with volume $V\left(M,{g}_{0}\right)=1$, smooth outside a compact set $\Sigma$, and is in ${W}_{loc}^{1,p}$ for some $p>n$. Suppose the scalar curvature of ${g}_{0}$ is at least $\sigma \left(M\right)$ 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 of the smooth structure. If $\Sigma$ is a compact embedded hypersurface, ${g}_{0}$ 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 a spin assumption we obtain a positive mass theorem on an asymptotically flat manifold for metrics with a compact singular set of codimension at least $2$.

##### Keywords
Yamabe invariants, positive mass theorems, singular metrics
Primary: 53C20
Secondary: 83C99