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

Mathematical Subject Classification 2010

Primary: 53C20

Secondary: 83C99

Milestones

Received: 1 December 2016

Revised: 4 May 2017

Accepted: 11 September 2017

Published: 23 November 2017