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

Abstract

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.

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