A lower bound for the radius of Weinstein's Lagrangian tubular neighborhood
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914158323171328 |
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| author | Yamamoto, Hikaru |
| author_facet | Yamamoto, Hikaru |
| contents | For an immersed Lagrangian submanifold $L$ in a Kähler manifold $(M,ω)$, there exists a symplectic local diffeomorphism from a tubular neighborhood of the image of the zero section in the normal bundle $T^{\bot}L$ of $L$, equipped with a canonical symplectic form $\tildeω$, to $(M,ω)$ whose restriction to $L$ is the identity map by Weinstein's Lagrangian tubular neighborhood theorem, where the image of the zero section in $T^{\bot}L$ is identified with $L$. In this paper, we give a lower bound for the supremum of the radii of tubular neighborhoods that have such a symplectic diffeomorphism into $(M,ω)$ from below by a constant explicitly given in terms of up to second derivatives of the Riemannian curvature tensor of $M$ and the second fundamental form of $L$. We also give a similar lower bound in the case where $L$ is compact and embedded. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10973 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A lower bound for the radius of Weinstein's Lagrangian tubular neighborhood Yamamoto, Hikaru Differential Geometry Symplectic Geometry 53C42, 53D12, 53C40, 53C21 For an immersed Lagrangian submanifold $L$ in a Kähler manifold $(M,ω)$, there exists a symplectic local diffeomorphism from a tubular neighborhood of the image of the zero section in the normal bundle $T^{\bot}L$ of $L$, equipped with a canonical symplectic form $\tildeω$, to $(M,ω)$ whose restriction to $L$ is the identity map by Weinstein's Lagrangian tubular neighborhood theorem, where the image of the zero section in $T^{\bot}L$ is identified with $L$. In this paper, we give a lower bound for the supremum of the radii of tubular neighborhoods that have such a symplectic diffeomorphism into $(M,ω)$ from below by a constant explicitly given in terms of up to second derivatives of the Riemannian curvature tensor of $M$ and the second fundamental form of $L$. We also give a similar lower bound in the case where $L$ is compact and embedded. |
| title | A lower bound for the radius of Weinstein's Lagrangian tubular neighborhood |
| topic | Differential Geometry Symplectic Geometry 53C42, 53D12, 53C40, 53C21 |
| url | https://arxiv.org/abs/2511.10973 |