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Estimating Reeb chords using microlocal sheaf theory

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arxiv 2106.04079 v3 pith:BXU2KSOD submitted 2021-06-08 math.SG math.GT

classification math.SGmath.GT
keywords chordslegendrianreebbettiboundhamiltonianlowermicrolocal
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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.

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  1. Weak Relative Calabi-Yau Structures for Legendrian Contact Homology

    math.SG 2025-09 conditional novelty 5.0 of 10

    For Legendrian knots in standard contact R3, the projection from the simply perturbed positive augmentation category to the circle category carries a weak right relative Calabi-Yau structure of dimension 2.

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