Complete Manifolds with Harmonic Curvature and Finite Lp-Norm Curvature

Let（M, g）(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L-norm of R?m is finite.As applications, we prove that（M, g） is compact if the L-norm of R?m is finite and R is positive, and（M, g） is scalar flat if（M, g） is a complete noncompact manifold with nonnegative scalar curvature and finite L-norm of R?m. We prove that（M, g） is isometric to a spherical space form if for p ≥n/2, the L-norm of R?m is sufficiently small and R is positive.In particular, we prove that（M, g） is isometric to a spherical space form if for p ≥ n, R is positive and the L-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.