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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.16187 |
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| _version_ | 1866914000908845056 |
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| author | Chen, Jiahuang |
| author_facet | Chen, Jiahuang |
| contents | We prove that for any nondegenerate $\mathbb{Z}/2$ harmonic $1$-form, there exists a metric perturbation producing a new nondegenerate $\mathbb{Z}/2$ harmonic $1$-form whose ordinary zero set is discrete. As an application, we show that for generic smooth nondegenerate $\mathbb{Z}/2$ harmonic $1$-forms, the leaf spaces are $\mathbb{Z}$-trees. Moreover, we show that if a $3$-dimensional rational homology sphere admits a smooth nondegenerate $\mathbb{Z}/2$-harmonic $1$-form, then there exists another nondegenerate $\mathbb{Z}/2$-harmonic $1$-form whose singular locus has exactly two connected components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perturbation and Pruning of Nondegenerate $\mathbb{Z}/2$ Harmonic 1-forms Chen, Jiahuang Differential Geometry We prove that for any nondegenerate $\mathbb{Z}/2$ harmonic $1$-form, there exists a metric perturbation producing a new nondegenerate $\mathbb{Z}/2$ harmonic $1$-form whose ordinary zero set is discrete. As an application, we show that for generic smooth nondegenerate $\mathbb{Z}/2$ harmonic $1$-forms, the leaf spaces are $\mathbb{Z}$-trees. Moreover, we show that if a $3$-dimensional rational homology sphere admits a smooth nondegenerate $\mathbb{Z}/2$-harmonic $1$-form, then there exists another nondegenerate $\mathbb{Z}/2$-harmonic $1$-form whose singular locus has exactly two connected components. |
| title | Perturbation and Pruning of Nondegenerate $\mathbb{Z}/2$ Harmonic 1-forms |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2508.16187 |