Lagrangian Surplusection Phenomena
Suppose you have a family of Lagrangian submanifolds ₜ and an auxiliary Lagrangian . Suppose that intersects some of the ₜ more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of ? Or will any Lagrangian is...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212783 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Lagrangian Surplusection Phenomena. Georgios Dimitroglou Rizell and Jonathan David Evans. SIGMA 20 (2024), 109, 13 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862692699515125760 |
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| author | Dimitroglou Rizell, Georgios Evans, Jonathan David |
| author_facet | Dimitroglou Rizell, Georgios Evans, Jonathan David |
| citation_txt | Lagrangian Surplusection Phenomena. Georgios Dimitroglou Rizell and Jonathan David Evans. SIGMA 20 (2024), 109, 13 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Suppose you have a family of Lagrangian submanifolds ₜ and an auxiliary Lagrangian . Suppose that intersects some of the ₜ more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of ? Or will any Lagrangian isotopic to surplusect some of the fibres? We argue that in several important situations, surplusection cannot be eliminated, and that a better understanding of surplusection phenomena (better bounds and a clearer understanding of how the surplusection is distributed in the family) would help to tackle some outstanding problems in different areas, including Oh's conjecture on the volume-minimising property of the Clifford torus and the concurrent normals conjecture in convex geometry. We pose many open questions.
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| first_indexed | 2026-03-17T23:35:33Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212783 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T23:35:33Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Dimitroglou Rizell, Georgios Evans, Jonathan David 2026-02-11T10:37:52Z 2024 Lagrangian Surplusection Phenomena. Georgios Dimitroglou Rizell and Jonathan David Evans. SIGMA 20 (2024), 109, 13 pages 1815-0659 2020 Mathematics Subject Classification: 53D12; 53D40 arXiv:2408.14883 https://nasplib.isofts.kiev.ua/handle/123456789/212783 https://doi.org/10.3842/SIGMA.2024.109 Suppose you have a family of Lagrangian submanifolds ₜ and an auxiliary Lagrangian . Suppose that intersects some of the ₜ more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of ? Or will any Lagrangian isotopic to surplusect some of the fibres? We argue that in several important situations, surplusection cannot be eliminated, and that a better understanding of surplusection phenomena (better bounds and a clearer understanding of how the surplusection is distributed in the family) would help to tackle some outstanding problems in different areas, including Oh's conjecture on the volume-minimising property of the Clifford torus and the concurrent normals conjecture in convex geometry. We pose many open questions. This note grew out of many conversations with many people, including Chris Evans, Ivan Smith, John Pardon, Umut Varolgunes, Leonid Polterovich, Kai Hugtenburg, Joé Brendel, Felix Schlenk, Elliot Gathercole, Nikolas Adaloglou, and Matt Buck. We are particularly grateful to Egor Shelukhin for pointing out the work of Viterbo [28], which led us to the short paper [12] of Goldstein, both of which give even more evidence for the ubiquity of surplusection (whilst also rendering our original volume bound obsolete). We also thank the referees for their prompt and insightful comments. The EPSRC Grant EP/W015749/1 supports J.E. G.D.R., which is supported by the Knut and Alice Wallenberg Foundation under grants KAW 2021.0191 and KAW 2021.0300, and by the Swedish Research Council under grant 2020-04426. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Lagrangian Surplusection Phenomena Article published earlier |
| spellingShingle | Lagrangian Surplusection Phenomena Dimitroglou Rizell, Georgios Evans, Jonathan David |
| title | Lagrangian Surplusection Phenomena |
| title_full | Lagrangian Surplusection Phenomena |
| title_fullStr | Lagrangian Surplusection Phenomena |
| title_full_unstemmed | Lagrangian Surplusection Phenomena |
| title_short | Lagrangian Surplusection Phenomena |
| title_sort | lagrangian surplusection phenomena |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212783 |
| work_keys_str_mv | AT dimitroglourizellgeorgios lagrangiansurplusectionphenomena AT evansjonathandavid lagrangiansurplusectionphenomena |