In this paper, aimed at exploring the fundamental properties of isoperimetric region in 3-manifold (M 3 , g) which is asymptotic to Anti-de Sitter-Schwarzschild manifold with scalar curvature R ≥ −6, we prove that connected isoperimetric region {D i } with H 3 g (D i ) ≥ δ 0 > 0 cannot slide off to the infinity of (M 3 , g) provided that (M 3 , g) is not isometric to the hyperbolic space. Furthermore, we prove that isoperimetric region {D i } with topological sphere {∂D i } as boundary is exhausting regions of M if Hawking mass m H (∂D i ) has uniform bound. In the case of exhausting isoperimetric region, we obtain a formula on expansion of isoperimetric profile in terms of renormalized volume.

In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature R ≥ −n (n − 1) and also the rigidity result when certain relative volume is zero.Resumé. Dans cet article, nous utilisons le flot de Ricci-DeTurk normalisé pour prouver la stabilité des variétés d'Einstein strictement stables et conformément compactes. En tant qu'application, nous montrons une comparaison de volume local pour les variétés conformément compactes dont la courbure scalaire satisfait R ≥ −n(n − 1). Nous donnonségalement un résultat de rigidité lorsque certain volume relative est nul.

No 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 ≥ 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.

No abstract