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

Sun-Yung Alice Chang; Yuxin Ge

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arXiv · 2509.18430

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

Sun-Yung Alice Chang·Yuxin Ge

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Abstract

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 $S^1 \times S^2$.

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BibTeX RIS

Sun-Yung Alice Chang, Yuxin Ge. 2025-09-22. On the problem of filling by a Poincar\'e-Einstein metric in dimension 4. https://arxiv.org/abs/2509.18430

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