A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound

We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions when the associated cone $\Gamma$ satisfies $\mu^+_\Gamma\le 1$, which includes the $\sigma_k-$Yamabe problem for $k$ not smaller than half of the dimension of the manifold.