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

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In this paper we show that for a generalized Berger metric g on S³ close to the round metric, the conformally compact Einstein (CCE) manifold (M, g) with (S³, [g]) as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if g is an SU(k+1)-invariant metric on S²ᵏ⁺¹ for k≥1, the non-positively curved CCE metric on the (2k+1)-ball B₁(0) with (S²ᵏ⁺¹, [g]) as its conformal infinity is unique up to isometries. In particular, since in LiQingShi, we proved that if the Yamabe constant of the conformal infinity Y(S²ᵏ⁺¹, [g]) is close to that of the round sphere then any CCE manifold filled in must be negatively curved and simply connected, therefore if g is an SU(k+1)-invariant metric on S²ᵏ⁺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 (S²ᵏ⁺¹, [g]) when the metric g is SU(k+1)-invariant.