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.