Advanced Nonlinear Studies (Mar 2024)

Perturbation compactness and uniqueness for a class of conformally compact Einstein manifolds

  • Chang Sun-Yung Alice,
  • Ge Yuxin,
  • Jin Xiaoshang,
  • Qing Jie

DOI
https://doi.org/10.1515/ans-2023-0124
Journal volume & issue
Vol. 24, no. 1
pp. 247 – 278

Abstract

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In this paper, we establish compactness results for some classes of conformally compact Einstein metrics defined on manifolds of dimension d ≥ 4. In the special case when the manifold is the Euclidean ball with the unit sphere as the conformal infinity, the existence of such class of metrics has been established in the earlier work of C. R. Graham and J. Lee (“Einstein metrics with prescribed conformal infinity on the ball,” Adv. Math., vol. 87, no. 2, pp. 186–225, 1991). As an application of our compactness result, we derive the uniqueness of the Graham–Lee metrics. As a second application, we also derive some gap theorem, or equivalently, some results of non-existence CCE fill-ins.

Keywords