Singular metrics with negative scalar curvature

Motivated by the work of Li and Mantoulidis, we study singular metrics which are uniformly Euclidean $(L^{\mathrm{\infty }})$ on a compact manifold $M^n$ ($n\ge 3$) with negative Yamabe invariant $\sigma (M)$. It is well-known that if $g$ is a smooth metric on $M$ with unit volume and with scalar curvature $R(g)\ge \sigma(M)$,then$g$ is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles $\le 2\pi$ along codimension-2 submanifolds. We also show in three dimension, if the Yamabe invariant of connected sum of two copies of $M$ attains its minimum, then the same is true for $L^{\mathrm{\infty }}$ metrics with isolated point singularities.