Singular metrics with negative scalar curvature

Abstract

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 [Formula: see text] on a compact manifold [Formula: see text] ([Formula: see text]) with negative Yamabe invariant [Formula: see text]. It is well known that if [Formula: see text] is a smooth metric on [Formula: see text] with unit volume and with scalar curvature [Formula: see text], then [Formula: see text] is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles [Formula: see text] along codimension-2 submanifolds. We also show in three dimensions, if the Yamabe invariant of connected sum of two copies of [Formula: see text] attains its minimum, then the same is true for [Formula: see text] metrics with isolated point singularities.