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

Romain Gicquaud (Laboratoire de Mathématiques et de Physique Théorique, UFR Sciences et Technologie, Université Franc¸ois Rabelais, Tours, France)

Dandan Ji (Key Laboratory of Pure and Applied Mathematics, School of Mathematics Science, Peking University, Beijing, China)

Yuguang Shi (Key Laboratory of Pure and Applied Mathematics, School of Mathematics Science, Peking University, Beijing, China)

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.

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