On the problem of filling by a Poincaré-Einstein metric in dimension 4
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912853876801536 |
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| author | Chang, Sun-Yung Alice Ge, Yuxin |
| author_facet | Chang, Sun-Yung Alice Ge, Yuxin |
| contents | Given a metric defined on a manifold of dimension three, we study the problem of finding a conformal filling by a Poincaré-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 $S^1 \times S^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18430 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the problem of filling by a Poincaré-Einstein metric in dimension 4 Chang, Sun-Yung Alice Ge, Yuxin Differential Geometry Given a metric defined on a manifold of dimension three, we study the problem of finding a conformal filling by a Poincaré-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 $S^1 \times S^2$. |
| title | On the problem of filling by a Poincaré-Einstein metric in dimension 4 |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.18430 |