I recall reading somewhere that if a conformal class contains an Einstein metric then that metric is the unique metric with constant scalar curvature in its conformal class, with the exception of the case of the round sphere. Does this sound right? If it is true: where can I find the proof of this result? 



Einstein metrics and conformal geometry
This is not the original reference, but the most general (i.e., applicable to the non-compact and pseudo-Riemannian cases) result I know is:


[Kühnel, W., & Rademacher, H. Conformal diffeomorphisms preserving the Ricci tensor. Proceedings of the American Mathematical Society, 123 (1995), no. 9, 2841–2848.](http://www.ams.org/mathscinet-getitem?mr=1260173)


(The [article](http://www.ams.org/journals/proc/1995-123-09/S0002-9939-1995-1260173-6/home.html) is publicly available for free.)


Theorem 1\* in there gives uniqueness in any case other than simply-connected, constant curvature spaces, and certain warped products with Ricci-flat spaces.


Check out the references in that paper, too.


