Small-energy isotopies of loose Legendrian submanifolds

Nakamura, Lukas¹

¹ Department of Mathematics, Uppsala University, Uppsala, Sweden

Geometry & topology, Tome 28 (2024) no. 9, pp. 4233-4255.

Résumé

We prove that for a closed Legendrian submanifold $L$ of dimension $n\ge 2$ 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 $\frac{1}{2}\eta$. 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 (2020).

Détail

DOI : 10.2140/gt.2024.28.4233

Keywords: Legendrian submanifolds, loose Legendrians, Shelukhin–Chekanov–Hofer energy, displacement of Legendrian submanifolds

Bibliographie

[1] A Banyaga, The structure of classical diffeomorphism groups, 400, Kluwer Acad. (1997) | DOI

[2] G Dimitroglou Rizell, M G Sullivan, The persistence of the Chekanov–Eliashberg algebra, Selecta Math. 26 (2020) 69 | DOI

[3] G Dimitroglou Rizell, M G Sullivan, The persistence of a relative Rabinowitz–Floer complex, preprint (2021)

[4] G Dimitroglou Rizell, M G Sullivan, C0–limits of Legendrians and positive loops, preprint (2022)

[5] H Geiges, An introduction to contact topology, 109, Cambridge Univ. Press (2008) | DOI

[6] E Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, PhD thesis, Stanford University (2012)

[7] Y G Oh, Geometry and analysis of contact instantons and entanglement of Legendrian links, I, preprint (2021)

[8] D Rosen, J Zhang, Chekanov’s dichotomy in contact topology, Math. Res. Lett. 27 (2020) 1165 | DOI

[9] E Shelukhin, The Hofer norm of a contactomorphism, J. Symplectic Geom. 15 (2017) 1173 | DOI