Lagrangian Surplusection Phenomena

Dimitroglou Rizell, Georgios and Evans, Jonny (2024) Lagrangian Surplusection Phenomena. SIGMA (Symmetry, Integrability and Geometry: Methods and Applications), 20: 109. ISSN 1815-0659

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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.