Poincaré-Einstein 4-manifolds with cusps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911715342417920 |
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| author | Li, Mingyang Liu, Hongyi |
| author_facet | Li, Mingyang Liu, Hongyi |
| contents | In this paper, we construct Poincaré-Einstein 4-manifolds with various kinds of cusps. In particular, we construct:
(1) Infinite families of Einstein metrics on $(0,\infty)\times \mathscr{N}$, where $\mathscr{N}\to T^2$ is a principal $\mathbb{S}^1$-bundle over $T^2$, with one Poincaré-Einstein end and one end asymptotic to a real or complex hyperbolic cusp.
(2) Infinite families of Einstein metrics on $(0,\infty)\times P$, where $P\to Σ_{\mathtt{g}}$ is a principal $\mathbb{S}^1$-bundle over a closed Riemann surface $Σ_{\mathtt{g}}$ of genus $\mathtt{g}\geq 2$, with one Poincaré-Einstein end and one end asymptotic to a bundle of two-dimensional hyperbolic cusps over hyperbolic $Σ_{\mathtt{g}}$.
Universal covers of (1) and (2) provide new complete negative Einstein metrics on $\mathbb{R}^4$. These Einstein metrics also exhibit interesting degeneration phenomena. With this construction, we give a negative answer to a question of Anderson concerning cusp formation for Poincaré-Einstein 4-manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25462 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Poincaré-Einstein 4-manifolds with cusps Li, Mingyang Liu, Hongyi Differential Geometry Mathematical Physics Analysis of PDEs In this paper, we construct Poincaré-Einstein 4-manifolds with various kinds of cusps. In particular, we construct: (1) Infinite families of Einstein metrics on $(0,\infty)\times \mathscr{N}$, where $\mathscr{N}\to T^2$ is a principal $\mathbb{S}^1$-bundle over $T^2$, with one Poincaré-Einstein end and one end asymptotic to a real or complex hyperbolic cusp. (2) Infinite families of Einstein metrics on $(0,\infty)\times P$, where $P\to Σ_{\mathtt{g}}$ is a principal $\mathbb{S}^1$-bundle over a closed Riemann surface $Σ_{\mathtt{g}}$ of genus $\mathtt{g}\geq 2$, with one Poincaré-Einstein end and one end asymptotic to a bundle of two-dimensional hyperbolic cusps over hyperbolic $Σ_{\mathtt{g}}$. Universal covers of (1) and (2) provide new complete negative Einstein metrics on $\mathbb{R}^4$. These Einstein metrics also exhibit interesting degeneration phenomena. With this construction, we give a negative answer to a question of Anderson concerning cusp formation for Poincaré-Einstein 4-manifolds. |
| title | Poincaré-Einstein 4-manifolds with cusps |
| topic | Differential Geometry Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2605.25462 |