Z/2 harmonic 1-forms, R-trees, and the Morgan-Shalen compactification
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909323752374272 |
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| author | He, Siqi Wentworth, Richard Zhang, Boyu |
| author_facet | He, Siqi Wentworth, Richard Zhang, Boyu |
| contents | This paper studies the relationship between an analytic compactification of the moduli space of flat $\mathrm{SL}_2(\mathbb{C})$ connections on a closed, oriented 3-manifold $M$ defined by Taubes, and the Morgan-Shalen compactification of the $\mathrm{SL}_2(\mathbb{C})$ character variety of the fundamental group of $M$. We exhibit an explicit correspondence between $\mathbb{Z}/2$ harmonic 1-forms, measured foliations, and equivariant harmonic maps to $\mathbb{R}$-trees, as initially proposed by Taubes. As an application, we prove that $\mathbb{Z}/2$ harmonic 1-forms exist on all Haken manifolds with respect to all Riemannian metrics. We also show that there exist manifolds that support singular $\mathbb{Z}/2$ harmonic 1-forms but have compact $\mathrm{SL}_2(\mathbb{C})$ character varieties, which resolves a folklore conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Z/2 harmonic 1-forms, R-trees, and the Morgan-Shalen compactification He, Siqi Wentworth, Richard Zhang, Boyu Differential Geometry Geometric Topology 58D27, 14M35, 57K35 This paper studies the relationship between an analytic compactification of the moduli space of flat $\mathrm{SL}_2(\mathbb{C})$ connections on a closed, oriented 3-manifold $M$ defined by Taubes, and the Morgan-Shalen compactification of the $\mathrm{SL}_2(\mathbb{C})$ character variety of the fundamental group of $M$. We exhibit an explicit correspondence between $\mathbb{Z}/2$ harmonic 1-forms, measured foliations, and equivariant harmonic maps to $\mathbb{R}$-trees, as initially proposed by Taubes. As an application, we prove that $\mathbb{Z}/2$ harmonic 1-forms exist on all Haken manifolds with respect to all Riemannian metrics. We also show that there exist manifolds that support singular $\mathbb{Z}/2$ harmonic 1-forms but have compact $\mathrm{SL}_2(\mathbb{C})$ character varieties, which resolves a folklore conjecture. |
| title | Z/2 harmonic 1-forms, R-trees, and the Morgan-Shalen compactification |
| topic | Differential Geometry Geometric Topology 58D27, 14M35, 57K35 |
| url | https://arxiv.org/abs/2409.04956 |