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# Communications in Analysis and Geometry

## Volume 21 (2013)

### Number 5

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

Pages: 1081 – 1113

DOI: http://dx.doi.org/10.4310/CAG.2013.v21.n5.a8

#### Authors

#### Abstract

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 $(X^n, g), n \geq 7$, admits an essential set, satisfies $\mathrm{Ric} = - (n - 1)g$, and has its Weyl curvature in $L^p$ for some $1 \lt p \lt \frac{n−1}{2}$, 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.