On the Poincaré-Einstein manifolds with cylindrical conformal infinity
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908556851150848 |
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| author | Chang, Sun-Yung Alice Yang, Paul Zhang, Ruobing |
| author_facet | Chang, Sun-Yung Alice Yang, Paul Zhang, Ruobing |
| contents | In this paper, we prove several rigidity and quantitative rigidity results for asymptotically hyperbolic Poincaré-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 Poincaré-Einstein metrics such as the AdS-Schwarzschild metrics on $D^2 \times S^{n - 1}$.
The main result shows that any Poincaré-Einstein filling of $S^1 \times S^{n - 1}$ must be hyperbolic if it is non-positively curved. As corollaries, the Poincaré-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 $ε$-regularity for the Weyl curvature of a general class of Poincaré-Einstein manifolds with conformal infinity of positive Yamabe type, which includes non-compact and volume-collapsed families of Poincaré-Einstein spaces in all dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20325 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Poincaré-Einstein manifolds with cylindrical conformal infinity Chang, Sun-Yung Alice Yang, Paul Zhang, Ruobing Differential Geometry In this paper, we prove several rigidity and quantitative rigidity results for asymptotically hyperbolic Poincaré-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 Poincaré-Einstein metrics such as the AdS-Schwarzschild metrics on $D^2 \times S^{n - 1}$. The main result shows that any Poincaré-Einstein filling of $S^1 \times S^{n - 1}$ must be hyperbolic if it is non-positively curved. As corollaries, the Poincaré-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 $ε$-regularity for the Weyl curvature of a general class of Poincaré-Einstein manifolds with conformal infinity of positive Yamabe type, which includes non-compact and volume-collapsed families of Poincaré-Einstein spaces in all dimensions. |
| title | On the Poincaré-Einstein manifolds with cylindrical conformal infinity |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.20325 |