Tightness of Chekanov's bound on displacement energy for some Lagrangian knots

TL;DR

This paper computes the displacement energy and minimal area of pseudo-holomorphic disks for certain Lagrangian tori, revealing cases where Chekanov's bound is tight and others where it is not, with implications for Lagrangian classification.

Contribution

It provides explicit calculations of displacement energy and minimal disk area for specific Lagrangian tori, demonstrating instances where Chekanov's bound is tight and where it is not.

Findings

For Chekanov tori in CP^n, e = ħ.

For exotic tori in C^3, e = ħ.

An example where e > ħ, showing Chekanov's bound is not always tight.

Abstract

By a classical theorem of Chekanov, the displacement energy, $e$, of a Lagrangian submanifold is bounded from below by the minimal area of pseudo-holomorphic disks with boundary on the Lagrangian, $\hbar$. We compute $e$ and $\hbar$ for displaceable Chekanov tori in $\mathbb{C}P^n$, and for an infinite family of exotic tori in $\mathbb{C}^3$ constructed by Brendel. In these families, $e=\hbar$. We compare continuity properties of $e$ and $\hbar$ on the space of Lagrangians. This provides an example (suggested by Fukaya, Oh, Ohta, and Ono) where $e>\hbar$. Our calculations have further applications such as a new proof, inspired by work of Auroux, that Brendel's family of exotic tori consists of infinitely many distinct Lagrangians.