On the problem of filling by a Poincar\'e-Einstein metric in dimension 4

Given a metric defined on a manifold of dimension three, we study the problem of finding a conformal filling by a Poincar\'e-Einstein metric on a manifold of dimension four. We establish a compactness result for classes of conformally compact Einstein $4$-manifolds under conformally invariant conditions. A key step in the proof is a result of rigidity for the hyperbolic metric on $\mathbb {B}^4$ or $S^1 \times \mathbb{B}^3$. As an application, we also derive some existence results of conformal filling in for metrics in a definite size neighborhood of the canonical metric; when the conformal infinity is either $S^3$ or $S^1 \times S^2$.