Lagrangian surplusection phenomena

Abstract

In this paper, we introduce a broad class of phenomena which appear when you intersect a given Lagrangian submanifold $K$ with a family of Lagrangian submanifolds $L_t$ (all Hamiltonian isotopic to one another). We establish that this phenomenon occurs in a particular situation, which lets us give a lower bound for the volume of any Lagrangian torus in $\mathbb{CP}^2$ which is Hamiltonian isotopic to the Chekanov torus. The rest of the paper is a discussion of why we should expect these phenomena to be very common, motivated by 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.