Flexible Lagrangians

Eliashberg, Yakov; Ganatra, Sheel; Lazarev, Oleg

doi:10.1093/imrn/rny078

Cited by 29 publications

(61 citation statements)

References 39 publications

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“…Second, a separate open question is whether every exact Lagrangian filling of a Legendrian Λ ⊆ (W 0 , λ 0 ) can be built from Legendrian isotopies and ambient surgeries. Lagrangians which can be built in this way are exactly those Lagrangians which are regular in the sense of [31]. It is an open question whether every Lagrangian in a Weinstein manifold is regular, and this is (if true) stronger than all previous questions: regular Lagrangians are exactly those which can be built from isotopies, ambient surgeries, and cores of ambient handles.…”

Section: 5mentioning

confidence: 99%

Legendrian fronts for affine varieties

Casals¹,

Murphy²

2019

Duke Math. J.

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In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed exact Lagrangian submanifolds. In addition, we prove that the Koras-Russell cubic is Stein deformation equivalent to C 3 and verify the affine parts of the algebraic mirrors of two Weinstein 4-manifolds.

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Section: 5mentioning

confidence: 99%

Legendrian fronts for affine varieties

Casals¹,

Murphy²

2019

Duke Math. J.

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“…For example, a relative analog of the Eliashberg-McDuff-Floer theorem states that any exact Lagrangian filling L n ⊂ B 2n of the standard Legendrian unknot in (S 2n−1 , ξ std ) is diffeomorphic to B n ; see [16], [1]. Recently, Eliashberg, Ganatra, and the author introduced the class of flexible Lagrangians [34]. These Lagrangians are the relative analog of flexible Weinstein domains.…”

Section: Flexiblementioning

confidence: 99%

Contact Manifolds with Flexible Fillings

Lazarev

2020

Geom. Funct. Anal.

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We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology; in certain cases, we prove that all flexible fillings are symplectomorphic. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. Similar methods are used to construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. We also prove relative analogs of our results, which we apply to Lagrangians in cotangent bundles.

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“…Let Σ(∂L) be the Weinstein thickening of the (possibly empty) Legendrian boundary ∂L. A Lagrangian L is called regular, see [18], if the Weinstein pair (X, Σ(∂L)) admits a skeleton which contains L. The problem is widely open. While no examples of non-regular Lagrangians are known, in the opposite direction in the case of a closed exact Lagrangian L in a general Weinstein domain X it is even unknown whether L realizes a non-zero homology class in H n (X) (which is a necessary condition for its regularity).…”

Section: Lagrangian Submanifolds Of Weinstein Domainsmentioning

confidence: 99%

“…If L ⊂ X is regular then by removing its tubular neighborhood N (L) one gets a Weinstein cobordism X L := (W \N (L), ∂ − X L := ∂N (L) \ ∂X, ∂ + X L := ∂X \ N (L)) (between manifolds with boundary if ∂L = ∅) whose negative boundary is the unit cotangent bundle of L. The Lagrangian L is called flexible, see [18]), if the cobordism X L is flexible.…”

Section: Lagrangian Submanifolds Of Weinstein Domainsmentioning

confidence: 99%

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Weinstein manifolds revisited

Eliashberg¹

2018

Proceedings of Symposia in Pure Mathematics

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This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds. Weinstein manifolds, domains, cobordismsWe begin with a notion of a Liouville domain. Let (X, ω) be a 2n-dimensional compact symplectic manifold with boundary equipped with an exact symplectic form ω. A Liouville structure on (X, ω) is a choice of a primitive λ, dλ = ω, called Liouville form such that λ| ∂X is a contact form and the orientation of ∂X by the form λ ∧ dλ n−1 | ∂X coincides with its orientation as the boundary of symplectic manifold (X, ω). The vector field Z, that is ω-dual to λ, i.e. ι(Z)ω = λ, is also called Liouville. It satisfies the condition L Z ω = ω which means that its flow is conformally symplectically expanding. The contact boundary condition is equivalent to the outward transversality of Z to ∂X. A Liouville domain X can always be completed to a Liouville manifold X by attaching a cylindrical end:and extending λ to X as equal to e s (λ| ∂X ) on the attached end. We will be constantly going back and forth between these two tightly related notions of Liouville domains and Liouville manifolds.Given a Liouville structure L = (X, ω, Z) we say that a Liouville structure L = (X , ω, Z) is obtained by a radial deformation from L if there exists a function h : *

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Flexible Lagrangians

Cited by 29 publications

References 39 publications

Legendrian fronts for affine varieties

Legendrian fronts for affine varieties

Contact Manifolds with Flexible Fillings

Weinstein manifolds revisited

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