Singular metrics with negative scalar curvature

Motivated by the work of Li and Mantoulidis [C. Li, C. Mantoulidis, Positive scalar curvature with skeleton singularities, Math. Ann. 374(1-2) (2019) 99-131], we study singular metrics which are uniformly Euclidean (L-infinity) on a compact manifold M-n (n >= 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 S(g) >= sigma(M), then g is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles <= 2 pi along codimension-2 submanifolds. We also show in three dimensions, if the Yamabe invariant of connected sum of two copies of M attains its minimum, then the same is true for L-infinity metrics with isolated point singularities.