Conformally flat Riemannian manifolds with finite $$L^p$$ L p -norm curvature

Abstract

Let \((M^n, g)(n\ge 3)\) 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 \(p\ge \frac{n}{2}\), 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 \(L^{p}(p\ge \frac{n}{2})\)-norm of T pinching theorem for complete, simply connected, locally conformally flat Riemannian manifolds with negative constant scalar curvature.