Explicit correspondences between gradient trees in $\mathbb{R}$ and holomorphic disks in $T^{*}\mathbb{R}$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914302216110080 |
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| author | Suzuki, Hidemasa |
| author_facet | Suzuki, Hidemasa |
| contents | Fukaya and Oh studied the correspondence between pseudoholomorphic disks in $T^{*}M$ which are bounded by Lagrangian sections $\{L_{i}^ε\}$ and gradient trees in $M$ which consist of gradient curves of $\{f_{i}-f_{j}\}$. Here, $L_{i}^ε$ is defined by $L_{i}^ε=$\,graph$(εdf_{i})$. They constructed approximate pseudoholomorphic disks in the case $ε>0$ is sufficiently small. When $M=\mathbb{R}$ and Lagrangian sections are affine, pseudoholomorphic disks $w_ε$ can be constructed explicitly. In this paper, we show that pseudoholomorphic disks $w_ε$ converges to the gradient tree in the limit $ε\to+0$ when the number of Lagrangian sections is three and four. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit correspondences between gradient trees in $\mathbb{R}$ and holomorphic disks in $T^{*}\mathbb{R}$ Suzuki, Hidemasa Symplectic Geometry Classical Analysis and ODEs Complex Variables Fukaya and Oh studied the correspondence between pseudoholomorphic disks in $T^{*}M$ which are bounded by Lagrangian sections $\{L_{i}^ε\}$ and gradient trees in $M$ which consist of gradient curves of $\{f_{i}-f_{j}\}$. Here, $L_{i}^ε$ is defined by $L_{i}^ε=$\,graph$(εdf_{i})$. They constructed approximate pseudoholomorphic disks in the case $ε>0$ is sufficiently small. When $M=\mathbb{R}$ and Lagrangian sections are affine, pseudoholomorphic disks $w_ε$ can be constructed explicitly. In this paper, we show that pseudoholomorphic disks $w_ε$ converges to the gradient tree in the limit $ε\to+0$ when the number of Lagrangian sections is three and four. |
| title | Explicit correspondences between gradient trees in $\mathbb{R}$ and holomorphic disks in $T^{*}\mathbb{R}$ |
| topic | Symplectic Geometry Classical Analysis and ODEs Complex Variables |
| url | https://arxiv.org/abs/2503.14080 |