A Lagrangian camel

Théret, David

doi:10.1007/s000140050107

Cited by 14 publications

(10 citation statements)

References 14 publications

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“…They have been used in numerous deep applications, such as metrics on infinite-dimensional groups of symmetries [Vit92, Sch00, Oh05b, Kha09, Lec08, San10, MZ11, CS12, Zap13a, Zap13b, Sey14]; the symplectic camel problem [Vit92,Thé99]; quasi-morphisms on the Hamiltonian group [EP03,Ost06,Ush11,FOOO11]; quasi-states and symplectic and contact rigidity [EP06, EP09, MVZ12, Zap13a]; orderability and contact nonsqueezing [San11,AM13]; C 0 -symplectic topology [Sey13a,Sey13b,HLS14,HLS15a,HLS15b]; function theory on symplectic manifolds [EPZ07,BEP12], [PR14] and the references therein; quantum measurements and noise [Pol12,Pol14]; surface dynamics [HLRS15]; and contact dynamics [Zap13a]. There are other applications, and it is not feasible to list all of them here, but the above sample should give the reader a feeling of the power of this wonderful tool of symplectic topology.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Spectral invariants for monotone Lagrangians

2018

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Abstract. Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [Vit92], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a "classical" Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Spectral invariants for monotone Lagrangians

2018

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“…By a classical result, see for example Appendix B of [17] or [18], the shifted index of the fiber function…”

Section: Proposition 53 Suppose λ Is Legendrian Isotopic Tomentioning

confidence: 99%

Generating function polynomials for legendrian links

Traynor

2001

Geom. Topol.

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It is shown that, in the 1-jet space of the circle, the swapping and the flyping procedures, which produce topologically equivalent links, can produce nonequivalent legendrian links. Each component of the links considered is legendrian isotopic to the 1-jet of the 0-function, and thus cannot be distinguished by the classical rotation number or Thurston-Bennequin invariants. The links are distinguished by calculating invariant polynomials defined via homology groups associated to the links through the theory of generating functions. The many calculations of these generating function polynomials support the belief that these polynomials carry the same information as a refined version of Chekanov's first order polynomials which are defined via the theory of holomorphic curves.

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“…, x n , y n ) maps ϕ − T (a) to ϕ + T (a) , and hence ϕ − T (a) ∼ ϕ + T (a) . However, if a is such that a ≥ c, then ϕ − T (a) ≈ ϕ + T (a) by the Lagrangian Camel Theorem of [25]. Therefore, the connectedness requirement cannot be omitted in Theorem 5.1.…”

Section: Spaces Of Symplectic Charts and Product Torimentioning

confidence: 99%

Lagrangian product tori in symplectic manifolds

Chekanov¹,

Schlenk²

2016

Comment. Math. Helv.

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In [3], product Lagrangian tori in standard symplectic space Ê 2n were classified up to symplectomorphism. We extend this classification to tame symplectically aspherical symplectic manifolds. We show by examples that the asphericity assumption cannot be omitted. Symplectic invariants2.1. Displacement energy and J-holomorphic discs. The first Ekeland-Hofer capacity was a key tool used in [3] for the classification of product tori in Ê 2n . This capacity is defined only for subsets of the standard symplectic space Ê 2n . We shall work with the displacement energy capacity instead, which is defined for all symplectic manifolds. In the process of computing the displacement energy for Lagrangian tori, we bring J-holomorphic discs into play, and it is here that we need the assumption that (M, ω) be tame.

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A Lagrangian camel

Abstract: Abstract. We prove the Lagrangian analogue of the symplectic camel theorem: there are compact Lagrangian submanifolds of R 2n that cannot be moved through a small hole by a global Hamiltonian isotopy with compact support.Mathematics Subject Classification (1991). 58F05.

Cited by 14 publications

References 14 publications

Spectral invariants for monotone Lagrangians

Spectral invariants for monotone Lagrangians

Generating function polynomials for legendrian links

Lagrangian product tori in symplectic manifolds

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