Scalar curvature and singular metrics

TL;DR

This paper proves that certain singular metrics on compact manifolds with nonpositive Yamabe invariant are Einstein outside their singular set, and establishes a positive mass theorem for asymptotically flat manifolds with singularities.

Contribution

It demonstrates that metrics with specific scalar curvature bounds and regularity conditions are Einstein outside singularities, extending regularity and positive mass results to singular settings.

Findings

Metrics with scalar curvature at least the Yamabe invariant are Einstein outside singularities.

Lipschitz continuous metrics are smooth and Einstein after changing the smooth structure.

A positive mass theorem holds for asymptotically flat manifolds with codimension ≥ 2 singular sets.

Abstract

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 $2$.