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

TL;DR

This paper establishes a compactness theorem for solutions to a fully nonlinear Yamabe problem under a lower Ricci curvature bound, leading to existence results for certain geometric equations on manifolds.

Contribution

It proves a new compactness result for nonlinear Yamabe solutions with Ricci bounds, extending existence results to a broader class of problems including the sigma_k-Yamabe case.

Findings

Solutions are compact under Ricci curvature bounds when the manifold is not conformally diffeomorphic to the sphere.

Existence of solutions is established for the problem when the cone parameter satisfies mma_\u2212_\u03b31.

Includes the sigma_k-Yamabe problem for k at least half the dimension of the manifold.

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.