Small energy isotopies of loose Legendrian submanifolds

Abstract

We prove that for a closed Legendrian submanifold L of dimension n > 2 with a loose chart of size η, any Legendrian isotopy starting at L can be C 0 -approximated by a Legendrian isotopy with energy arbitrarily close to η 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 [DRS20].