In this paper we show that for an $\text{ Sp}(k+1)$ invariant metric $\hat{ g}$ on $\mathbb{ S}^{ 4k+3}$ $(k\geq 1)$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(\mathbb{ S}^{ 4k+3}, [\hat{ g}])$ as its conformal infinity is unique up to isometries. Moreover, by the result in [LiQingShi], $g$ is the Graham-Lee metric on the unit ball $B_1\subset \mathbb{ R}^{ 4k+4}$. We also give an a priori estimate on the Einstein metric $g$. Based on the estimate and Graham-Lee and Lee's seminal perturbation result, we use the continuity method directly to obtain an existence result of the non-positively curved CCE metric with prescribed conformal infinity $(\mathbb{ S}^{ 4k+3}, [\hat{ g}])$ when the metric $\hat{ g}$ is $\text{ Sp}(k+1)$-invariant. We also generalize the results to the case of conformal infinity $(\mathbb{ S}^{ 15},[\hat{ g}])$ with $\hat{ g}$ a Spin$(9)$-invariant metric in the appendix.
Categories: math.DG