A sharp inequality relating Yamabe invariants on asymptotically Poincaré–Einstein manifolds with a Ricci curvature lower bound

Let [Formula: see text] be a conformally compact manifold with [Formula: see text]. If [Formula: see text] is asymptotically Poincaré–Einstein, we establish a sharp inequality relating the type II Yamabe invariant of [Formula: see text] and the Yamabe invariant of its conformal infinity.