Rigidity of complete noncompact bach-flat nn-manifolds

Let (Mn,g)(Mn,g) be a complete noncompact Bach-flat nn-manifold with the positive Yamabe constant and constant scalar curvature. Assume that the L2L2-norm of the trace-free Riemannian curvature tensor R∘m is finite. In this paper, we prove that (Mn,g)(Mn,g) is a constant curvature space if the Ln2-norm of R∘m is sufficiently small. Moreover, we get a gap theorem for (Mn,g)(Mn,g) with positive scalar curvature. This can be viewed as a generalization of our earlier results of 4-dimensional Bach-flat manifolds with constant scalar curvature R≥0R≥0 [Y.W. Chu, A rigidity theorem for complete noncompact Bach-flat manifolds, J. Geom. Phys. 61 (2011) 516–521]. Furthermore, when n>9n>9, we derive a rigidity result for R<0R<0.