Abstract: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 $P\to \Sigma_{\mathtt{g}}$ is a principal $\mathbb{S}^1$-bundle over a closed Riemann surface $\Sigma_{\mathtt{g}}$ 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 $\Sigma_{\mathtt{g}}$.
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.
| Comments: | 35 pages. Comments are welcome |
| Subjects: | Differential Geometry (math.DG); Mathematical Physics (math-ph); Analysis of PDEs (math.AP) |
| Cite as: | arXiv:2605.25462 [math.DG] |
| (or arXiv:2605.25462v1 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25462 arXiv-issued DOI via DataCite (pending registration) |
From: Hongyi Liu [view email]
[v1]
Mon, 25 May 2026 06:14:19 UTC (41 KB)
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