Abstract: A manifold is called Poincaré-Einstein if it has constant negative Ric curvature and admits a suitable compactification. As the framework for AdS/CFT correspondence, it has been intensively studied over the last thirty years. One guiding principle in this area is to understand the connection between the manifold and its conformal boundary, and in my talk I will show an inequality between Yamabe invariants in this direction and then generalize this to manifolds with a lower Ric curvature bound.