Conformally flat Riemannian manifolds with finite LP-norm curvature

Let (M-n , g)(n >= 3) be an n-dimensional complete, simply connected, locally conformally flat Riemannian manifold with constant scalar curvature S. Denote by T the trace-free Ricci curvature tensor of M. The main result of this paper states that T goes to zero uniformly at infinity if for p >= n/2, the L-P-norm of T is finite. As applications, we prove that (M-n , g) is compact if the L-P-norm of T is finite and S is positive, and (M-n , g) is scalar flat if (M-n , g) is a noncompact manifold with nonnegative constant scalar curvature and the L-P-norm of T is finite. We prove that (M-n , g) is isometric to a sphere if S is positive and the L-P-norm of T is pinched in [0, C), where C is an explicit positive constant depending only on n, p and S. Finally, we prove an L-P (p >= n/2)-norm of T pinching theorem for complete, simply connected, locally conformally flat Riemannian manifolds with negative constant scalar curvature.