Cohomogeneity one $\mathrm{Spin}(7)$ metrics with generic Aloff--Wallach spaces as principal orbits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915693492961280 |
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| author | Chi, Hanci |
| author_facet | Chi, Hanci |
| contents | This paper establishes the existence of forward complete cohomogeneity one $\mathrm{Spin}(7)$ metrics with generic Aloff--Wallach spaces $N_{k,l}$ as principal orbits and $\mathbb{CP}^2$ as the singular orbit, building on Reidegeld's analysis of the initial value problem. We construct three continuous one-parameter families of non-compact $\mathrm{Spin}(7)$ metrics. Each family contains a limiting asymptotically conical (AC) metric, while the other metrics in the families are asymptotically locally conical (ALC). Moreover, two of the AC metrics share the same asymptotic cone, exhibiting a geometric transition phenomenon analogous to that found by Lehmann in the exceptional case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21267 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomogeneity one $\mathrm{Spin}(7)$ metrics with generic Aloff--Wallach spaces as principal orbits Chi, Hanci Differential Geometry 53C25 This paper establishes the existence of forward complete cohomogeneity one $\mathrm{Spin}(7)$ metrics with generic Aloff--Wallach spaces $N_{k,l}$ as principal orbits and $\mathbb{CP}^2$ as the singular orbit, building on Reidegeld's analysis of the initial value problem. We construct three continuous one-parameter families of non-compact $\mathrm{Spin}(7)$ metrics. Each family contains a limiting asymptotically conical (AC) metric, while the other metrics in the families are asymptotically locally conical (ALC). Moreover, two of the AC metrics share the same asymptotic cone, exhibiting a geometric transition phenomenon analogous to that found by Lehmann in the exceptional case. |
| title | Cohomogeneity one $\mathrm{Spin}(7)$ metrics with generic Aloff--Wallach spaces as principal orbits |
| topic | Differential Geometry 53C25 |
| url | https://arxiv.org/abs/2512.21267 |