On Uniqueness of Conformally Compact Einstein Metrics with Homogeneous Conformal Infinity

Abstract

In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3,[\hat{g}])$ as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \cite{LiQingShi}, we proved that if the Yamabe constant of the conformal infinity $Y(S^3,[\hat{g}])$ 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 $\hat{g}$ is a Berger metric on $S^3$ with $Y(S^3,[\hat{g}])$ close to that of the round metric, the conformally compact Einstein metric filled in is unique up to isometries.

Some changes are made in the references. Comments are welcome!

References in corpus (1)

• Gap phenomena and curvature estimates for Conformally Compact Einstein Manifolds

Cited by in corpus (1)

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