2017/12/18 by Gang Li, Li, Gang
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1712.06215
openalex publication_date 2017/12/18 · openalex created_date 2018/01/05 · openalex updated_date 2026/07/28
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 \citeLiQingShi, 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.