Scalar curvature and singular metrics

Abstract

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 $V(M,g_0)=1$, 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 $\sigma (M)$ 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 $2$.

47pages, All comments are welcome

References in corpus (2)

• Kähler-Ricci flow with unbounded curvature
• A positive mass theorem for low-regularity Riemannian metrics

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