An equation of Monge-Ampere type in conformal geometry, and   four-manifolds of positive Ricci curvature
  We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched. 
Sun-Yung A. Chang, Matthew J. Gursky and Paul C. Yang