  In this paper, we study scalar curvature rigidity of non-smooth metrics on
smooth manifolds with non-positive Yamabe invariant. We prove that if the
scalar curvature is not less than the Yamabe invariant in distributional sense,
then the manifold must be isometric to an Einstein manifold. This result
extends Theorem 1.4 in Jiang, Sheng and the first author (Sci. China Math. 66
(2023) no. 6, 1141-1160), from a special case where the manifolds have zero
Yamabe invariant to general cases where the manifolds have non-positive Yamabe
invariant. This result depends highly on an analysis and estimates of geometric
evolution equations.
