Small energy isotopies of loose Legendrian submanifolds

TL;DR

This paper establishes that loose Legendrian submanifolds can be approximated by isotopies with arbitrarily small energy, bounding their displacement energy and confirming a conjecture related to loose Legendrian displaceability.

Contribution

It proves a bound on the energy of Legendrian isotopies for loose Legendrians, confirming a conjecture and linking loose chart size to displacement energy.

Findings

Displacement energy of loose Legendrians is bounded by half the size of the loose chart.

Any Legendrian isotopy can be approximated by one with energy close to half the loose chart size.

The result confirms a conjecture by Dimitroglou Rizell and Sullivan.

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.