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

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, $\mathrm{\hslash }$. We compute $e$ and $\mathrm{\hslash }$ 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, $e=\mathrm{\hslash }$. We compare continuity properties of $e$ and $\mathrm{\hslash }$ on the space of Lagrangians. This provides an example (suggested by Fukaya, Oh, Ohta, and Ono) where $e>\mathrm{\hslash }$. 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.