On locally conformally flat manifolds with positive pinched Ricci curvature

Abstract

By using the Yamabe flow, we prove that if $(M^n,g)$, $n\geq3$, is an $n$-dimensional locally conformally flat complete Riemannian manifold $Rc\geq \epsilon Rg>0$, where $\epsilon>0$ is a uniformly constant, then $M^n$ must be compact. Our result shows that Hamilton's pinching conjecture also holds for higher dimensional case if we assume additionally the metric is locally conformally flat.