On uniqueness of conformally compact Einstein metrics with homogeneous conformal infinity

Abstract

In this paper we show that for a Berger metric gˆ on S3, the non-positively curved conformally compact Einstein metric on the 4-ball B1(0) with (S3,[gˆ]) as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen [21]. In particular, since in [18], we proved that if the Yamabe constant of the conformal infinity Y(S3,[gˆ]) is close to that of the round sphere then any conformally compact Einstein manifold filled in must be negatively curved and simply connected, therefore if gˆ is a Berger metric on S3 with Y(S3,[gˆ]) close to that of the round metric, the conformally compact Einstein metric filled in is unique up to isometries.