A Construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n}$
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908590310162432 |
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| author | Yan, Dashen |
| author_facet | Yan, Dashen |
| contents | We discover an explicit construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n},n\geq 3$, using a variant of ellipsoidal coordinates on $\mathbb{R}^{n}$. The branching set of these examples is a codimension-$2$ ellipsoid, providing the first known family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ with compact branching sets. Moreover, the graph of the related $\mathbb{Z}_{2}$-harmonic one form in $T^{*}\mathbb{R}^{n}$ can be obtained as a certain limit of a specific sequence of Lawlor's necks in $\mathbb{C}^{n}=T^{*}\mathbb{R}^{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n}$ Yan, Dashen Differential Geometry Classical Analysis and ODEs We discover an explicit construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n},n\geq 3$, using a variant of ellipsoidal coordinates on $\mathbb{R}^{n}$. The branching set of these examples is a codimension-$2$ ellipsoid, providing the first known family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ with compact branching sets. Moreover, the graph of the related $\mathbb{Z}_{2}$-harmonic one form in $T^{*}\mathbb{R}^{n}$ can be obtained as a certain limit of a specific sequence of Lawlor's necks in $\mathbb{C}^{n}=T^{*}\mathbb{R}^{n}$. |
| title | A Construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n}$ |
| topic | Differential Geometry Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.19286 |