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

TL;DR

This paper investigates the asymptotic geometry of Einstein manifolds with Weyl curvature in certain L^p spaces, showing they are asymptotically hyperbolic under specific conditions and establishing a rigidity result for hyperbolic space.

Contribution

It proves that Einstein manifolds with Weyl curvature in L^p spaces are asymptotically locally hyperbolic, extending understanding of curvature decay and geometric structure at infinity.

Findings

Manifolds with Weyl curvature in L^p are asymptotically hyperbolic.

Established a rigidity result for hyperbolic space under integral curvature conditions.

Provided conditions for the geometric behavior near infinity of Einstein manifolds.

Abstract

In this paper 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 and 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 of this theorem is to show a rigidity result for the hyperbolic space under an integral condition for the curvature.