Free boundary Hamiltonian stationary Lagrangian discs in $\mathbb{C}^2$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916343061676032 |
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| author | Gaia, Filippo |
| author_facet | Gaia, Filippo |
| contents | Let $Ω\subset \mathbb{C}^2$ be a smooth domain. We establish conditions under which a weakly conformal, branched $Ω$-free boundary Hamiltonian stationary Lagrangian immersion $u$ of a disc in $\mathbb{C}^2$ is a $Ω$-free boundary minimal immersion. We deduce that if $u$ is a weakly conformal, branched $B_1(0)$-free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary data, then $u(D^2)$ must be a Lagrangian equatorial plane disc. We also present examples of $Ω$-free boundary Hamiltonain stationary discs, demonstrating the optimality of our assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Free boundary Hamiltonian stationary Lagrangian discs in $\mathbb{C}^2$ Gaia, Filippo Differential Geometry Analysis of PDEs 53D12, 53C42, 53C24, 58E12 Let $Ω\subset \mathbb{C}^2$ be a smooth domain. We establish conditions under which a weakly conformal, branched $Ω$-free boundary Hamiltonian stationary Lagrangian immersion $u$ of a disc in $\mathbb{C}^2$ is a $Ω$-free boundary minimal immersion. We deduce that if $u$ is a weakly conformal, branched $B_1(0)$-free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary data, then $u(D^2)$ must be a Lagrangian equatorial plane disc. We also present examples of $Ω$-free boundary Hamiltonain stationary discs, demonstrating the optimality of our assumptions. |
| title | Free boundary Hamiltonian stationary Lagrangian discs in $\mathbb{C}^2$ |
| topic | Differential Geometry Analysis of PDEs 53D12, 53C42, 53C24, 58E12 |
| url | https://arxiv.org/abs/2408.00748 |