2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67382In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to $B^4$ and its conformal infinity is diffeomorphic to $S^3$.16 pagesDifferential GeometryAnalysis of PDEs53C25, 53C80, 58J60text