On Uniqueness And Existence of Conformally Compact Einstein Metrics with Homogeneous Conformal Infinity

In this paper we show that for a generalized Berger metric $\hat{g}$ on $S^3$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if $\hat{g}$ is an $\text{SU}(k+1)$-invariant metric on $S^{2k+1}$ for $k\geq1$, the non-positively curved CCE metric on the $(2k+1)$-ball $B_1(0)$ with $(S^{2k+1}, [\hat{g}])$ as its conformal infinity is unique up to isometries. In particular, since in \cite{LiQingShi}, we proved that if the Yamabe constant of the conformal infinity $Y(S^{2k+1}, [\hat{g}])$ is close to that of the round sphere then any CCE manifold filled in must be negatively curved and simply connected, therefore if $\hat{g}$ is an $\text{SU}(k+1)$-invariant metric on $S^{2k+1}$ which is close to the round metric, the CCE metric filled in is unique up to isometries. Using the continuity method, we prove an existence result of the non-positively curved CCE metric with prescribed conformal infinity $(S^{2k+1}, [\hat{g}])$ when the metric $\hat{g}$ is $\text{SU}(k+1)$-invariant.