Hofer’s symplectic energy and lagrangian intersections in contact geometry

Akaho, Manabu

doi:10.1215/kjm/1250517619

Cited by 6 publications

(5 citation statements)

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“…Given a choice of contact form α for (Y, ξ), we will associate quantitative measurements to Legendrian isotopies, modeled on the measurements introduced by [She14] for contactomorphisms. Similar quantitative measurements have been studied by multiple authors, see for instance [RZ20, Ush21, Hed21, Oh21, DRS20a, DRS20b, DRS21, DRS22b, Aka01,Her07]. The measurements we consider can be considered as Legendrian versions of the Hofer energy, [Hof90], for exact isotopies of Lagrangians, as in [Pol93,Che98,Che00].…”

Section: Introductionmentioning

confidence: 65%

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Remarks on the oscillation energy of Legendrian isotopies

Cant¹

2023

Preprint

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Section: Introductionmentioning

confidence: 65%

“…The proofs are based on two Floer homology theories counting intersection points between SΛ 0 and L in SY . In similar settings, the papers [Aka01], [Her07] prove persistence results which incorporate the oscillation energy.…”

Section: Introductionmentioning

confidence: 88%

Remarks on the oscillation energy of Legendrian isotopies

Cant¹

2023

Preprint

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“…For a Legendrian submanifold of a general contact manifold (again under the assumption that α = α − and Λ = Λ − ), Theorem 1.3 can also be seen to follow from a result by Akaho [3], again under the additional assumption that φ R Rα (Λ) ⊂ Y is a closed submanifold satisfying some additional topological constraints. Note that the latter behavior is non-generic and imposes severe restrictions on the contact form.…”

Section: 22mentioning

confidence: 85%

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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“…max |K(t, •)|d t < ħ h, thus proving the lemma modulo Claim 3.7. Claim 3.7 follows by arguments like those used in [EHS95],[Ak01],[DS16]; indeed the only difference between our situation and that of [DS16, Section 4] is that we are working with one compact Lagrangian and one cylindrical Lagrangian instead of two cylindrical Lagrangians. Suppose for contradiction that we had ũm ∈K c m ,R m (for all m ∈ + ) such that there does not exist any fixed compact set containing the image of every ũm .…”

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confidence: 87%

“…Our argument will follow the proof of [U14, Theorem 4.9] which establishes a similar statement for pairs of compact Lagrangian submanifolds of geometrically bounded symplectic manifolds. See also [Oh97] (in the symplectic case) and [Ak01] (in the contact case) for earlier related work. As we will see, the assumption that Λ is hypertight allows the proof to go through even though symplectizations are not geometrically bounded.…”

Section: Hypertightness and Local Rigidity For Legendriansmentioning

confidence: 99%

Local rigidity, contact homeomorphisms, and conformal factors

Usher¹

2020

Preprint

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We show that if the image of a Legendrian submanifold under a contact homeomorphism (i.e. a homeomorphism that is a C 0 -limit of contactomorphisms) is smooth then it is Legendrian, assuming only positive local lower bounds on the conformal factors of the approximating contactomorphisms. More generally the analogous result holds for coisotropic submanifolds in the sense of [H15]. This is a contact version of the Humilière-Leclercq-Seyfaddini coisotropic rigidity theorem in C 0 symplectic geometry, and the proof adapts the author's recent re-proof of that result in [U19] based on a notion of local rigidity of points on locally closed subsets. We also provide two different flavors of examples showing that a contact homeomorphism can map a submanifold that is transverse to the contact structure to one that is smooth and tangent to the contact structure at a point.

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Hofer’s symplectic energy and lagrangian intersections in contact geometry

Cited by 6 publications

References 2 publications

Remarks on the oscillation energy of Legendrian isotopies

Remarks on the oscillation energy of Legendrian isotopies

An Energy–Capacity inequality for Legendrian submanifolds

Local rigidity, contact homeomorphisms, and conformal factors

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