首页|On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity.Ⅱ
On uniqueness and existence of conformally compact Einstein metrics with homogeneous conformal infinity.Ⅱ
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In this paper,we show that for an Sp(k+1)-invariant metric g on S4k+3(k≥1)close to the round metric,the conformally compact Einstein(CCE)manifold(M,g)with(S4k+3,[(g)])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 B1 ⊂ R4k+4.We also give an a priori estimate of the Einstein metric g.As a by-product 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(S4k+3,[g])when the metric g is Sp(k+1)-invariant.We also generalize the results to the case of conformal infinity(S15,[g])with g a Spin(9)-invariant metric in the appendix.
conformally compact Einstein manifoldsprescribed conformal infinityuniqueness and existence of CCE filling-intwo-point boundary value problem of nonlinear ODE systemsvolume comparisontotal variation
Gang Li
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School of Mathematics,Shandong University,Jinan 250100,China
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