Extremal Lagrangian tori in toric domains
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914610179735552 |
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| author | Faisal, Shah |
| author_facet | Faisal, Shah |
| contents | Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,ω)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal Lagrangian torus in $(X,ω)$ is a Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in $(X,ω)$. We prove that every extremal Lagrangian torus in the symplectic unit ball $(\bar{B}^{2n}(1),ω_{\mathrm{std}})$ is contained entirely in the boundary $\partial B^{2n}(1)$. This answers a question attributed to Lazzarini and completely settles a conjecture of Cieliebak and Mohnke in the affirmative. In addition, we prove the conjecture for a class of toric domains in $(\mathbb{C}^n, ω_{\mathrm{std}})$, which includes all compact strictly convex four-dimensional toric domains. We explain with counterexamples that the general conjecture does not hold for non-convex domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13076 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal Lagrangian tori in toric domains Faisal, Shah Symplectic Geometry Differential Geometry Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,ω)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal Lagrangian torus in $(X,ω)$ is a Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in $(X,ω)$. We prove that every extremal Lagrangian torus in the symplectic unit ball $(\bar{B}^{2n}(1),ω_{\mathrm{std}})$ is contained entirely in the boundary $\partial B^{2n}(1)$. This answers a question attributed to Lazzarini and completely settles a conjecture of Cieliebak and Mohnke in the affirmative. In addition, we prove the conjecture for a class of toric domains in $(\mathbb{C}^n, ω_{\mathrm{std}})$, which includes all compact strictly convex four-dimensional toric domains. We explain with counterexamples that the general conjecture does not hold for non-convex domains. |
| title | Extremal Lagrangian tori in toric domains |
| topic | Symplectic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2504.13076 |