Estimating Reeb chords using microlocal sheaf theory
Abstract
We show that for a closed Legendrian submanifold in a 1-jet bundle, if there is a sheaf with compact support, perfect stalk and singular support on that Legendrian, then (1) the number of Reeb chords has a lower bound by half of the sum of Betti numbers of the Legendrian; (2) the number of Reeb chords between the original Legendrian and its Hamiltonian pushoff has a lower bound in terms of Betti numbers when the oscillation norm of the Hamiltonian is small comparing with the length of Reeb chords. In the proof we develop a duality exact triangle and use the persistence structure (which comes from the action filtration) of microlocal sheaves.
Keywords
Cite
@article{arxiv.2106.04079,
title = {Estimating Reeb chords using microlocal sheaf theory},
author = {Wenyuan Li},
journal= {arXiv preprint arXiv:2106.04079},
year = {2025}
}
Comments
39 pages, 8 figures. Comments are welcome! Added Rem 1.26 on relation with other works; modified Def 3.18; simplified proofs of Thm 4.7, 4.9, 4.11, Cor 4.12 & Prop 6.1; removed expository proofs of known results in Sec 5.2; added Rem 6.14 on the assumption; added Appendix A of non-horizontally displaceable Legendrians with sheaves but no generating functions. To appear in Algebr. Geom. Topol