A remark on the rigidity of Poincaré–Einstein manifolds

Abstract

In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{\'e}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the only conformally compact Poincar{\'e}-Einstein manifold with the round sphere as its conformal infinity.