A rigidity theorem for complete noncompact Bach-flat manifolds

Let (M⁴,g)$(M^4,g)$ be a four-dimensional complete noncompact Bach-flat Riemannian manifold with positive Yamabe constant. In this paper, we show that (M⁴,g)$(M^4,g)$ has a constant curvature if it has a nonnegative constant scalar curvature and sufficiently small _L2$L_2$-norm of trace-free Riemannian curvature tensor. Moreover, we get a gap theorem for (M⁴,g)$(M^4,g)$ with positive scalar curvature.