Small energy isotopies of loose Legendrian submanifolds

Abstract

We prove that for a closed Legendrian submanifold $L$ of dimension $n \geq 2$ with a loose chart of size $\eta$, any Legendrian isotopy starting at $L$ can be $C^0$-approximated by a Legendrian isotopy with energy arbitrarily close to $\frac{\eta}{2}$. This in particular implies that the displacement energy of loose displaceable Legendrians is bounded by half the size of its smallest loose chart, which proves a conjecture of Dimitroglou Rizell and Sullivan.