Einstein manifolds with finite Lp-norm of the Weyl curvature

Abstract

Let (Mn,g)(n≥4) be an n-dimensional complete Einstein manifold. Denote by W the Weyl curvature tensor of M. We prove that (Mn,g) is isometric to a spherical space form if (Mn,g) has positive scalar curvature and unit volume, and the Lp(p≥n2)-norm of W is pinched in [0,C), where C is an explicit positive constant depending only on n, p and S, which improves the isolation theorems given by [24,14,17].This paper also states that W goes to zero uniformly at infinity if for p≥n2, the Lp-norm of W of M with non-positive scalar curvature and positive Yamabe constant is finite. Assume that M has negative scalar curvature and the Lα-norm of W is finite. As application, we prove that M is a hyperbolic space form if the Lp-norm of W is sufficiently small, which generalizes an Ln2-norm of W pinching theorem in [19].