Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 109, 13 pages      arXiv:2408.14883      https://doi.org/10.3842/SIGMA.2024.109

Lagrangian Surplusection Phenomena

Georgios Dimitroglou Rizell a and Jonathan David Evans b
a) Department of Mathematics, Uppsala Universitet, Uppsala, Sweden
b) Department of Mathematics and Statistics, Lancaster University, Bailrigg, UK

Received September 03, 2024, in final form November 23, 2024; Published online December 06, 2024

Abstract
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.

Key words: symplectic geometry; Lagrangian intersections; Floer theory.

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