A remark on the rigidity of Poincaré–Einstein manifolds

In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poincaré–Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof (without any additional assumption) of the rigidity of the hyperbolic space as the only conformally compact Poincaré–Einstein manifold with the round sphere as its conformal infinity.