Lagrangian split tori in $S^2 \times S^2$ and billiards
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915139903553536 |
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| author | Brendel, Joé Kim, Joontae |
| author_facet | Brendel, Joé Kim, Joontae |
| contents | In this paper, we classify up to Hamiltonian isotopy Lagrangian tori that split as a product of circles in $S^2 \times S^2$, when the latter is equipped with a non-monotone split symplectic form. We show that this classification is equivalent to a problem of mathematical billiards in rectangles. We give many applications, among others: (1) answering a question on Lagrangian packing numbers raised by Polterovich--Shelukhin, (2) studying the topology of the space of Lagrangian tori, and (3) determining which split tori are images under symplectic ball embeddings of Chekanov or product tori in $\mathbb{R}^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lagrangian split tori in $S^2 \times S^2$ and billiards Brendel, Joé Kim, Joontae Symplectic Geometry Dynamical Systems 53D12 (Primary) 53D20 (Secondary) In this paper, we classify up to Hamiltonian isotopy Lagrangian tori that split as a product of circles in $S^2 \times S^2$, when the latter is equipped with a non-monotone split symplectic form. We show that this classification is equivalent to a problem of mathematical billiards in rectangles. We give many applications, among others: (1) answering a question on Lagrangian packing numbers raised by Polterovich--Shelukhin, (2) studying the topology of the space of Lagrangian tori, and (3) determining which split tori are images under symplectic ball embeddings of Chekanov or product tori in $\mathbb{R}^4$. |
| title | Lagrangian split tori in $S^2 \times S^2$ and billiards |
| topic | Symplectic Geometry Dynamical Systems 53D12 (Primary) 53D20 (Secondary) |
| url | https://arxiv.org/abs/2502.03324 |