Remarks on the oscillation energy of Legendrian isotopies

Abstract: We construct non-compact contact manifolds containing compact Legendrians which can be displaced from their Reeb flow with arbitrarily small oscillation energy. We use this to show the Shelukhin-Chekanov-Hofer pseudo-metric considered by Rosen and Zhang is degenerate on the isotopy class of the constructed Legendrians. Other aspects related to oscillation energy of Legendrian isotopies are explored.