On the Poincar\'e-Einstein manifolds with cylindrical conformal infinity

In this paper, we prove several rigidity and quantitative rigidity results for asymptotically hyperbolic Poincar\'e-Einstein manifolds whose conformal infinities are diffeomorphic to a cylinder $S^1 \times S^{n - 1}$. It is a basic fact that the Riemannian product $S^1 \times S^{n - 1}$ can bound, in addition to a complete hyperbolic metric on $S^1 \times D^n$, other Poincar\'e-Einstein metrics such as the AdS-Schwarzschild metrics on $D^2 \times S^{n - 1}$. The main result shows that any Poincar\'e-Einstein filling of $S^1 \times S^{n - 1}$ must be hyperbolic if it is non-positively curved. As corollaries, the Poincar\'e-Einstein filling of $S^1 \times S^{n - 1}$ is unique when the length of circle factor is sufficiently large or the $L^2$-energy of the Weyl curvature is sufficiently small relative to the Yamabe constant of the conformal infinity. To prove the Weyl pinching rigidity, we established a new $\epsilon$-regularity for the Weyl curvature of a general class of Poincar\'e-Einstein manifolds with conformal infinity of positive Yamabe type, which includes non-compact and volume-collapsed families of Poincar\'e-Einstein spaces in all dimensions.