On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity.Ⅱ  

作　　者：Gang Li 

机构地区：[1]School of Mathematics,Shandong University,Jinan,250100,China

出　　处：《Science China Mathematics》2024年第12期2789-2822,共34页中国科学（数学英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11701326);the Fundamental Research Funds of Shandong University(Grant No.2016HW008);the Young Scholars Program of Shandong University(Grant No.2018WLJH85)。

摘　　要：In this paper,we show that for an Sp(k+1)-invariant metric g on S^(4k+3)(k 1)close to the round metric,the conformally compact Einstein(CCE)manifold(M,g)with(S^(4k+3),[?])as its conformal infinity is unique up to isometry.Moreover,by the result in Li et al.(2017),g is the Graham-Lee metric(see Graham and Lee(1991))on the unit ball B_(1)■R^(4k+4).We also give an a priori estimate of the Einstein metric g.As a byproduct of the a priori estimates,based on the estimate and Graham-Lee and Lee's seminal perturbation results(see Graham and Lee(1991)and Lee(2006)),we directly use the continuity method to obtain an existence result of the non-positively curved CCE metric with prescribed conformal infinity(S^(4k+3),[g])when the metric?is Sp(k+1)-invariant.We also generalize the results to the case of conformal infinity(S^(15),[?])with g a Spin(9)-invariant metric in the appendix.

关 键 词：conformally compact Einstein manifolds prescribed conformal infinity uniqueness and existence of CCE filling-in two-point boundary value problem of nonlinear ODE systems volume comparison total variation 

分 类 号：O186[理学—数学]