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

Abstract

In this paper we show that for an $\text{Sp}(k+1)$ invariant metric $\hat{g}$ on ${\mathbb{S}}^{4k+3}$ $(k\ge 1)$ 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.

Final version, accepted for publication in SCIENCE CHINA Mathematics. arXiv admin note: substantial text overlap with arXiv:1712.06215

References in corpus (2)

• Rigidity of Asymptotically Hyperboblic Manifolds
• On Uniqueness And Existence of Conformally Compact Einstein Metrics with Homogeneous Conformal Infinity

Cited by in corpus (1)

• A construction of Poincaré-Einstein metrics of cohomogeneity one on the ball