Complete noncompact manifolds with harmonic curvature

Let (M (n) , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M (n) , g) is a space form if it has sufficiently small L (n/2)-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M (n) , g) with positive scalar curvature.