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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2308.05086 |
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| _version_ | 1866915895655268352 |
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| author | Li, Yin |
| author_facet | Li, Yin |
| contents | Given a closed, oriented Lagrangian submanifold $L$ in a Liouville domain $\overline{M}$, one can define a Maurer-Cartan element with respect to a certain $L_\infty$-structure on the string homology $\widehat{H}_\ast^{S^1}(\mathcal{L}L;\mathbb{R})$, completed with respect to the action filtration. When the first Gutt-Hutchings capacity of $\overline{M}$ is finite, and $L$ is a $K(π,1)$ space, we show that $L$ bounds a pseudoholomorphic disc of Maslov index 2. This confirms a general form of Audin's conjecture and generalizes the works of Fukaya and Irie in the case of $\mathbb{C}^n$ to a wide class of Liouville manifolds, which includes low degree smooth affine hypersurfaces in $\mathbb{C}^{n+1}$. In particular, when $\dim_\mathbb{R}(\overline{M})=6$, every closed, orientable, prime Lagrangian 3-manifold $L\subset\overline{M}$ is diffeomorphic either to a spherical space form, or $S^1\timesΣ_g$, where $Σ_g$ is a closed oriented surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05086 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations Li, Yin Symplectic Geometry Given a closed, oriented Lagrangian submanifold $L$ in a Liouville domain $\overline{M}$, one can define a Maurer-Cartan element with respect to a certain $L_\infty$-structure on the string homology $\widehat{H}_\ast^{S^1}(\mathcal{L}L;\mathbb{R})$, completed with respect to the action filtration. When the first Gutt-Hutchings capacity of $\overline{M}$ is finite, and $L$ is a $K(π,1)$ space, we show that $L$ bounds a pseudoholomorphic disc of Maslov index 2. This confirms a general form of Audin's conjecture and generalizes the works of Fukaya and Irie in the case of $\mathbb{C}^n$ to a wide class of Liouville manifolds, which includes low degree smooth affine hypersurfaces in $\mathbb{C}^{n+1}$. In particular, when $\dim_\mathbb{R}(\overline{M})=6$, every closed, orientable, prime Lagrangian 3-manifold $L\subset\overline{M}$ is diffeomorphic either to a spherical space form, or $S^1\timesΣ_g$, where $Σ_g$ is a closed oriented surface. |
| title | Aspherical Lagrangian submanifolds, Audin's conjecture and cyclic dilations |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2308.05086 |