Singular metrics with negative scalar curvature

Abstract

Motivated by the work of Li and Mantoulidis [14], we study singular metrics which are uniformly Euclidean (L ∞ ) on a compact manifold M n (n ≥ 3) with negative Yamabe invariant σ(M ). It is well-known that if g is a smooth metric on M with unit volume and with scalar curvature S(g) ≥ σ(M ), then g is Einstein. We show, in all dimensions, the same is true for metrics with edge singularities with cone angles ≤ 2π 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 ∞ metrics with isolated point singularities.