New examples of Z/2 harmonic 1-forms and their deformations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916604994912256 |
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| author | Haydys, Andriy Mazzeo, Rafe Takahashi, Ryosuke |
| author_facet | Haydys, Andriy Mazzeo, Rafe Takahashi, Ryosuke |
| contents | We collect a number of elementary constructions of $\Z_2$ harmonic $1$-forms, and of families of these objects. These examples show that the branching set $Σ$ of a $\Z_2$ harmonic 1-form may exhibit the following features: i) $Σ$ may be a non-trivial link; ii) $Σ$ may be a multiple cover; iii) $Σ$ may be immersed, and appear as a limit of smoothly embedded branching loci; iv) there are families of $\Z_2$ harmonic $1$-forms whose branching sets $Σ$ have tangent cones filling out a positive dimensional space, even modulo isometries. We show that Features i) and ii) occur already in dimension three, while the remaining ones appear at least in dimension four and higher. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06227 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New examples of Z/2 harmonic 1-forms and their deformations Haydys, Andriy Mazzeo, Rafe Takahashi, Ryosuke Differential Geometry Analysis of PDEs We collect a number of elementary constructions of $\Z_2$ harmonic $1$-forms, and of families of these objects. These examples show that the branching set $Σ$ of a $\Z_2$ harmonic 1-form may exhibit the following features: i) $Σ$ may be a non-trivial link; ii) $Σ$ may be a multiple cover; iii) $Σ$ may be immersed, and appear as a limit of smoothly embedded branching loci; iv) there are families of $\Z_2$ harmonic $1$-forms whose branching sets $Σ$ have tangent cones filling out a positive dimensional space, even modulo isometries. We show that Features i) and ii) occur already in dimension three, while the remaining ones appear at least in dimension four and higher. |
| title | New examples of Z/2 harmonic 1-forms and their deformations |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2307.06227 |