# Three-Dimensional Metrics as Deformations of a Constant Curvature Metric?

I read the following paper Three-Dimensional Metrics as Deformations of a Constant Curvature Metric and discovered the following result:

I have three questions:

(1) Is $h$ also a conformally flat metric?

(2) Is $\boldsymbol{s}$ strictly the gradient of a scalar function, the curl of another field, or both?

(3) Am I safe to assume that the proof of this theorem is valid? I was unable to find a peer reviewed equivalent.

Regarding (1) a three manifold is conformally flat if and only if the Cotton tensor vanishes. The Cotton tensor is defined as a linear combination of covariant derivatives of the curvature tensor, and hence, in particular, always vanishes for constant curvature metrics. So yes, $h$ is by definition conformally flat.
Regarding (2) a quick scan of the paper reveals no statement regarding whether $s$ can be taken to be either exact or co-exact, and I don't see a priori any reason why this should be true. (But I have no time to read the paper in detail.)