On nondegenerate $\mathbb{Z}_{2}$-harmonic $1$-forms with shrinking branching sets
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915869261561856 |
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| author | Yan, Dashen |
| author_facet | Yan, Dashen |
| contents | We develop a gluing theorem for non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on compact manifolds, in which non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ are glued to the regular zeros of a non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-form. As an immediate consequence, viewing an ordinary harmonic $1$-form as a $\mathbb{Z}_{2}$-harmonic $1$-form without branching set, we prove that for every compact oriented manifold $M^{n}, n\geq 3$, if the first Betti number $b^{1}(M)>0$, then $M$ admits a family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms, which resolves a folklore conjecture. We will also discuss several possible applications to special holonomy, in particular, to the field of $G_{2}$-geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On nondegenerate $\mathbb{Z}_{2}$-harmonic $1$-forms with shrinking branching sets Yan, Dashen Differential Geometry We develop a gluing theorem for non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on compact manifolds, in which non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ are glued to the regular zeros of a non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-form. As an immediate consequence, viewing an ordinary harmonic $1$-form as a $\mathbb{Z}_{2}$-harmonic $1$-form without branching set, we prove that for every compact oriented manifold $M^{n}, n\geq 3$, if the first Betti number $b^{1}(M)>0$, then $M$ admits a family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms, which resolves a folklore conjecture. We will also discuss several possible applications to special holonomy, in particular, to the field of $G_{2}$-geometry. |
| title | On nondegenerate $\mathbb{Z}_{2}$-harmonic $1$-forms with shrinking branching sets |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2510.07678 |