A formal Riemannian structure on conformal classes and uniqueness for   the $\sigma_{2}$-Yamabe problem
  We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the $\sigma_2$-Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere. 
Matthew J. Gursky, Jeffrey Streets
math.DG math.AP