We will take the conformal class of the Berger sphere as an example to show that when the conformal infinity is the conformal class of a homogeneous metric on the sphere which is close to the round metric, then the conformally compact Einstein metric that fills in is unique up to isometry.
Written by: Wu Chaochao
Edited by: Hu Rong
Source: Peking University Lecture Hall
http://lecture.pku.edu.cn/information.php?id=16063