On the asymptotic behavior of Einstein manifolds with an integral bound on the Weyl curvature

In this article, we consider the geometric behavior near infinity of some Einstein manifolds (X n , g) with Weyl curvature belonging to a certain L p space.Namely, we show that if (X n , g), n ≥ 7, admits an essential set, satisfies Ric = -(n -1)g, and has its Weyl curvature in L p for some 1 < p < n-1 2 , then the norm of the Weyl tensor decays exponentially fast at infinity.One interesting application of this theorem is to show a rigidity result for the hyperbolic space under an integral condition for the curvature.