A rigidity theorem for complete noncompact Bach-flat manifolds

Abstract

Let ( M 4 , g ) be a four-dimensional complete noncompact Bach-flat Riemannian manifold with positive Yamabe constant. In this paper, we show that ( M 4 , g ) has a constant curvature if it has a nonnegative constant scalar curvature and sufficiently small L 2 -norm of trace-free Riemannian curvature tensor. Moreover, we get a gap theorem for ( M 4 , g ) with positive scalar curvature.