Lagrangian Surplusection Phenomena

TL;DR

This paper investigates the phenomenon of surplus intersections in families of Lagrangian submanifolds, exploring conditions under which surplus intersections can or cannot be eliminated via Hamiltonian isotopies, with implications for symplectic topology and convex geometry.

Contribution

It provides new insights into the limitations of eliminating surplus intersections in Lagrangian families and highlights open problems in symplectic topology and related fields.

Findings

Surplus intersections often cannot be eliminated in several key situations.

Understanding surplusection distribution aids in tackling conjectures like Oh's conjecture and the normals conjecture.

The paper raises open questions about the bounds and distribution of surplusection phenomena.

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.