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| Format: | Preprint |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2002.05403 |
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| _version_ | 1866913554498584576 |
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| author | Mettler, Thomas |
| author_facet | Mettler, Thomas |
| contents | We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $\mathfrak{p}$ and a volume form $σ$ on an oriented surface $M$ equip the total space of a certain disk bundle $Z\to M$ with a pair $(J_{\mathfrak{p}},\mathfrak{J}_{\mathfrak{p},σ})$ of almost complex structures. A conformal structure on $M$ corresponds to a section of $Z\to M$ and $\mathfrak{p}$ is metrisable by the metric $g$ if and only if $[g] : M \to Z$ is a pseudo-holomorphic curve with respect to $J_{\mathfrak{p}}$ and $\mathfrak{J}_{\mathfrak{p},dA_g}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_05403 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Metrisability of projective surfaces and pseudo-holomorphic curves Mettler, Thomas Differential Geometry Symplectic Geometry We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure $\mathfrak{p}$ and a volume form $σ$ on an oriented surface $M$ equip the total space of a certain disk bundle $Z\to M$ with a pair $(J_{\mathfrak{p}},\mathfrak{J}_{\mathfrak{p},σ})$ of almost complex structures. A conformal structure on $M$ corresponds to a section of $Z\to M$ and $\mathfrak{p}$ is metrisable by the metric $g$ if and only if $[g] : M \to Z$ is a pseudo-holomorphic curve with respect to $J_{\mathfrak{p}}$ and $\mathfrak{J}_{\mathfrak{p},dA_g}$. |
| title | Metrisability of projective surfaces and pseudo-holomorphic curves |
| topic | Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2002.05403 |