Rigidity of complete noncompact Riemannian manifolds with harmonic curvature

For complete noncompact Riemannian manifolds (Mn,g) with harmonic curvature, we prove that g is Einstein under an inequality involving Ln2-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant. Furthermore, we achieve that Mn is a constant curvature space under such inequality and finite L2-norm of the Weyl curvature.