Rigidity of Schouten tensor under conformal deformation

We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng [5] recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition (\(Ric-\epsilon Rg\ge 0\)) must be compact. This answers higher dimensional Hamilton’s pinching conjecture on locally conformally flat manifolds affirmatively. Since (modified) Schouten tensor being nonnegative is equivalent to a Ricci pinching condition, our main result yields a simple proof of Cheng’s theorem.