Complete noncompact manifolds with harmonic curvature

Abstract

Let (Mn, 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 (Mn, g) is a space form if it has sufficiently small Ln/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (Mn, g) with positive scalar curvature.