On the incompleteness of $G_2$-moduli spaces along degenerating families of $G_2$-manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909697663041536 |
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| author | Langlais, Thibault |
| author_facet | Langlais, Thibault |
| contents | We derive a formula for the energy of a path in the moduli space of a compact $G_2$-manifold with vanishing first Betti number for the volume-normalised $L^2$-metric. This allows us to give simple sufficient conditions for a path of torsion-free $G_2$-structures to have finite energy and length. We deduce that the compact $G_2$-manifolds produced by the generalised Kummer construction have incomplete moduli spaces. Under some assumptions, we also state a necessary condition for the limit of a path of torsion-free $G_2$-structures to be at infinite distance in the moduli space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the incompleteness of $G_2$-moduli spaces along degenerating families of $G_2$-manifolds Langlais, Thibault Differential Geometry 53C26 (Primary), 53C29, 53A15, 58D27 (Secondary) We derive a formula for the energy of a path in the moduli space of a compact $G_2$-manifold with vanishing first Betti number for the volume-normalised $L^2$-metric. This allows us to give simple sufficient conditions for a path of torsion-free $G_2$-structures to have finite energy and length. We deduce that the compact $G_2$-manifolds produced by the generalised Kummer construction have incomplete moduli spaces. Under some assumptions, we also state a necessary condition for the limit of a path of torsion-free $G_2$-structures to be at infinite distance in the moduli space. |
| title | On the incompleteness of $G_2$-moduli spaces along degenerating families of $G_2$-manifolds |
| topic | Differential Geometry 53C26 (Primary), 53C29, 53A15, 58D27 (Secondary) |
| url | https://arxiv.org/abs/2405.02943 |