The minimal length of a Lagrangian cobordism between Legendrians

Sabloff, Joshua M.; Traynor, Lisa

doi:10.1007/s00029-016-0288-0

Cited by 7 publications

(5 citation statements)

References 39 publications

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“…Legendrian knots in contact 3-manifolds are instrumental to study the contact geometry of 3-manifolds [6,20,21,22,24,25,32]. The classification of Legendrian knots and their Lagrangian fillings has been one of the central areas of research in low-dimensional contact topology [17,18,36,47,49,50,51]. The only Legendrian knot for which there exist a complete non-empty classification of Lagrangian fillings is the Legendrian unknot [19].…”

Section: Introductionmentioning

confidence: 99%

Infinitely many Lagrangian fillings

Casals¹,

Gao²

2020

Preprint

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We prove that all maximal-tb Legendrian torus links (n, m) in the standard contact 3-sphere, except for (2, m), (3, 3), (3,4) and (3, 5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances which induce faithful actions of the modular group PSL(2, Z) and the mapping class group M0,4 into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.

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Section: Introductionmentioning

confidence: 99%

Infinitely many Lagrangian fillings

Casals¹,

Gao²

2020

Preprint

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“…Results related to Corollary 1.9. Sabloff and Traynor prove a similar result when the Legendrian contact homology DGAs have augmentations [47], which in turned inspired this corollary. Their hypotheses are stronger as many Legendrian contact homology DGAs do not have augmentations.…”

Section: 22mentioning

confidence: 57%

An Energy–Capacity inequality for Legendrian submanifolds

Rizell

Sullivan

2018

J. Topol. Anal.

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We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the Z2-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the established invariance enables us to use two different contact forms: one for the count of Reeb chords and another for the measure of the smallest length, under the assumption that there is a suitable symplectic cobordism from the latter to the former. The size of the contact isotopy is measured in terms of the oscillation of the contact Hamiltonian, together with the maximal factor by which the contact form is shrunk during the isotopy. The main tool used is a Mayer-Vietoris sequence for Lagrangian Floer homology, obtained by "neck-stretching" and "splashing." 1 4 GEORGIOS DIMITROGLOU RIZELL AND MICHAEL G. SULLIVAN Corollary 1.7. Let Λ ⊂ (Y, ξ) be a Legendrian submanifold and let α 0 be a (not necessarily generic) contact form on (Y, ξ) which is relatively hypertight, i.e. for which σ(α 0 , Λ) = +∞. Suppose that Λ ′ is Legendrian isotopic to Λ. For any choice of contact form α, we then have the boundgiven that the latter chords are transverse.Proof. The contact form α can be written as α = e f α 0 for some real-valued function f :After a sufficiently small perturbation of the contact form α − it may be assumed to be generic, while the inequality e A H osc < σ(α − , Λ) is still satisfied for the contact Hamiltonian generating the isotopy from Λ to φ 1 α,Ht (Λ). Since it is possible to assume that α − = e g α holds for some smooth function g : Y → (0, 1) also after the perturbation, the result now follows directly from Corollary 1.6.Example 1.8. The following are well-known examples of Legendrian submanifolds satisfying σ(α 0 , Λ) = +∞ to which the above corollary can be applied.(1) The zero section of the one-jet space (J 1 M, dz − λ M ) of a closed manifold M endowed with its canonical contact form α std = dz − λ M ; i.e. λ M is the Liouville form on T * M and z is the canonical coordinate on the R-factor of J 1 M = T * M × R;(2) A fiber of the unit cotangent bundle S * M ⊂ T * M with contact form α 0 = λ M | T (S * M ) induced by a Riemannian metric on M having non-positive sectional curvature (recall that such a Riemannian manifold has no closed contractible geodesics, even without assuming periodicity); and(3) The conormal lift of a sub-torus (S 1 ) k × {1} n−k ⊂ (S 1 ) n inside the unit cotangent bundle S * (S 1 ) n , with contact form α 0 induced by the canonical flat metric on (S 1 ) n = R n /Z n .In another direction, define the α-displacement energy of a Legendrian Λ in (Y, kerα) to be disp(α, Λ) := inf Ht e A H osc where the infimum is taken over all contact Hamiltonians such that there are no α-Reeb chords between φ 1 α,Ht (Λ) and Λ or vice versa. (Set this infimum to +∞ if no such Hamiltonian exists.) Suppose Λ − , Λ + ⊂ Y are two Legendrians which are Lagrangian concordant in the symplectization (R×Y, ...

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“…They are now a wellestablished technique for studying quantitative questions in symplectic topology. They have also been defined for Legendrian submanifolds in certain contact manifolds by Zapolsky in [52], and Sabloff-Traynor considered some of their properties under Lagrangian cobordisms in their work [46]. Here we study further properties that are satisfied under Lagrangian cobordisms and positive isotopies which will be used when proving Theorem 1.8.…”

Section: 4mentioning

confidence: 92%

Positive Legendrian isotopies and Floer theory

Chantraine¹,

Colin²,

Rizell³

2019

Annales De l'Institut Fourier

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Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds filled by a Liouville domain with non-zero symplectic homology are strongly orderable in the sense of Liu.

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The minimal length of a Lagrangian cobordism between Legendrians

Cited by 7 publications

References 39 publications

Infinitely many Lagrangian fillings

Infinitely many Lagrangian fillings

An Energy–Capacity inequality for Legendrian submanifolds

Positive Legendrian isotopies and Floer theory

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