A rigidity theorem for complete noncompact Bach-flat manifolds

Let (M4,g)(M4,g) be a four-dimensional complete noncompact Bach-flat Riemannian manifold with positive Yamabe constant. In this paper, we show that (M4,g)(M4,g) has a constant curvature if it has a nonnegative constant scalar curvature and sufficiently small L2L2-norm of trace-free Riemannian curvature tensor. Moreover, we get a gap theorem for (M4,g)(M4,g) with positive scalar curvature.