On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity

Abstract

In this paper we show that for a generalized Berger metric ${\hat{g}}$ on ${\mathbb {S}}^3$ close to the round metric, the conformally compact Einstein (CCE) manifold (M, g) with $({\mathbb {S}}^3, [{\hat{g}}])$ 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 ${\mathbb {S}}^{2k+1}$ for $k\ge 1$, the non-positively curved CCE metric on the $(2k+1)$-ball $B_1(0)$ with $({\mathbb {S}}^{2k+1}, [{\hat{g}}])$ as its conformal infinity is unique up to isometries. In particular, since in Li (Trans Amer Math Soc 369(6): 4385–4413, 2017), we proved that if the Yamabe constant of the conformal infinity $Y({\mathbb {S}}^{2k+1}, [{\hat{g}}])$ 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 ${\mathbb {S}}^{2k+1}$ 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 $({\mathbb {S}}^{2k+1}, [{\hat{g}}])$ when the metric ${\hat{g}}$ is $\text {SU}(k+1)$-invariant.