If one starts with a pair $(M,g)$ where $M$ is a Riemannian $3$-manifold and $g$ is the metric and further assumes that $g$ is a conformally flat metric with zero scalar curvature, if one then adds a conformally flat perturbation to the metric where the size of the perturbation is small in some suitable norm, will the perturbed metric necessarily have zero or positive scalar curvature, or can it also have negative scalar curvature?
As an example of a metric and such a perturbation, consider a metric which a perturbation of the spatial Schwarzchild metric (which has zero scalar curvature and is conformal to the Euclidean metric)
$g_{ij} = \Bigg( 1 + \frac{M}{2r} \Bigg)^4 \delta_{ij} + h_{ij} .$
In the case of a conformally flat perturbation, the perturbation does not push the metric away from its conformally flat form so it can be written as
$ g_{ij} = \Bigg( \Bigg( 1 + \frac{M}{2r} \Bigg)^4 + \sigma \Bigg) \delta_{ij},$
where $\sigma$ is a function of the spatial coordinates.