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In this paper we consider the geometric behavior near infinity of someEinstein manifolds $(X^n, g)$ with Weyl curvature belonging to a certain $L^p$space. Namely, we show that if $(X^n, g)$, $n \geq 7$, admits an essential setand has its Weyl curvature in $L^p$ for some $1<p<\frac{n-1}{2}$, then $(X^n,g)$ must be asymptotically locally hyperbolic. One interesting application ofthis theorem is to show a rigidity result for the hyperbolic space under anintegral condition for the curvature.

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