Final published version, 483 KB, PDF document
Available under license: CC BY-SA: Creative Commons Attribution-ShareAlike 4.0 International License
Final published version
Licence: CC BY-SA: Creative Commons Attribution-ShareAlike 4.0 International License
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Lagrangian Surplusection Phenomena
AU - Dimitroglou Rizell, Georgios
AU - Evans, Jonny
PY - 2024/12/6
Y1 - 2024/12/6
N2 - Suppose you have a family of Lagrangian submanifolds L_t and an auxiliary Lagrangian K. Suppose that K intersects some of the L_t more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of K? Or will any Lagrangian isotopic to K 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.
AB - Suppose you have a family of Lagrangian submanifolds L_t and an auxiliary Lagrangian K. Suppose that K intersects some of the L_t more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of K? Or will any Lagrangian isotopic to K 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.
U2 - 10.3842/SIGMA.2024.109
DO - 10.3842/SIGMA.2024.109
M3 - Journal article
VL - 20
JO - SIGMA (Symmetry, Integrability and Geometry: Methods and Applications)
JF - SIGMA (Symmetry, Integrability and Geometry: Methods and Applications)
SN - 1815-0659
M1 - 109
ER -