Legendrians with vanishing Shelukhin–Chekanov–Hofer metric

Abstract

We show that the Legendrian lift of an exact, displaceable Lagrangian has vanishing Shelukhin–Chekanov–Hofer pseudo-metric by lifting an argument due to Sikorav to the contactization. In particular, this proves the existence of such Legendrians, providing counterexamples to a conjecture of Rosen and Zhang.