A note on the uniqueness of solutions for the Yamabe problem
  We prove that in conformal classes of metrics near the class of an Einstein metric (other than the standard round metric on a sphere) the Yamabe problem has a unique solution up to scaling. This is a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata. 
L. L. de Lima, P. Piccione, M. Zedda
math.DG