Displacement of polydisks and Lagrangian Floer theory

Fukaya, Kenji; Oh, Yong─Geun; Ohta, Hiroshi; Ono, Kaoru

doi:10.4310/jsg.2013.v11.n2.a4

Cited by 24 publications

(40 citation statements)

References 12 publications

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“…When one unravels the arguments underlying the proofs, though, it becomes clear that this is not really a new proof of Chekanov's theorem, as similar ideas (though organized differently, of course) have been used in proofs such as the one in [CL05, Section 4.3]. This approach to estimating the displacement energy of a Lagrangian submanifold seems to be very closely related to the approach using "torsion thresholds" in Floer homology in [FOOO11].…”

Section: Introductionmentioning

confidence: 98%

Hofer’s metrics and boundary depth

Usher¹

2013

Ann. Sci. École Norm. Sup.

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We show that if (M , ω) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of (M , ω) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer's metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in M × M when M satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theoretic quantity called the boundary depth, which measures the nontriviality of the boundary operator on the Floer complex in a way that encodes robust symplectic-topological information.1991 Mathematics Subject Classification. 53D22, 53D40.

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Section: Introductionmentioning

confidence: 98%

Hofer’s metrics and boundary depth

Usher¹

2013

Ann. Sci. École Norm. Sup.

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“…See [13]. We now have corrected it in [13]. Then the proof of Theorem 15.7 is the same as that of Theorem J given in [13].…”

Section: 2mentioning

confidence: 95%

“…We now have corrected it in [13]. Then the proof of Theorem 15.7 is the same as that of Theorem J given in [13]. Theorem 6.1.25 [6] can be generalized in the same way.…”

Section: 2mentioning

confidence: 98%

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Lagrangian floer Theory over integers: spherically positive symplectic manifolds

Fukaya

Ohta

et al. 2013

Pure and Applied Mathematics Quarterly

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Abstract. In this paper we study the Lagrangian Floer theory over Z or Z 2 . Under an appropriate assumption on ambient symplectic manifold, we show that the whole story of Lagrangian Floer theory in [6], [7] can be developed over Z 2 coefficients, and over Z coefficients when Lagrangian submanifolds are relatively spin. The main technical tools used for the construction are the notion of the sheaf of groups, and stratification and compatibility of the normal cones applied to the Kuranishi structure of the moduli space of pseudoholomorphic discs.

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“…We refer to section 5.3 [FOOO1] or section 5 [FOOO3] for the details of the construction of the Floer-Piunikhin (pre)-chain map Ψ H .…”

Section: Thick-thin Dichotomy Of Floer Trajectoriesmentioning

confidence: 99%

Localization of Floer homology of engulfed topological Hamiltonian loop

2013

Communications in Information and Systems

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Localization of Floer homology is first introduced by Floer [Fl3]in the context of Hamiltonian Floer homology. The author employed the notion in the Lagrangian context for the pair (φ 1 H (L), L) of compact Lagrangian submanifolds in tame symplectic manifolds (M, ω) in [Oh1, Oh2] for a compact Lagrangian submanifold L and C 2 -small Hamiltonian H. In this article, motivated by the study of topological Hamiltonian dynamics, we extend the localization process for any engulfable Hamiltonian path φ H whose time-one map φ 1 H is sufficiently C 0 -close to the identity (and also to the case of triangle product), and prove that the value of local Lagrangian spectral invariant is the same as that of global one. Such a Hamiltonian path naturally occurs as an approximating sequence of engulfable topological Hamiltonian loop. We also apply this localization to the graphs Graph φ t H in (M × M, ω ⊕ −ω) and localize the Hamiltonian Floer complex of such a Hamiltonian H. We expect that this study will play an important role in the study of homotopy invariance of the spectral invariants of topological Hamiltonian.

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Displacement of polydisks and Lagrangian Floer theory

Cited by 24 publications

References 12 publications

Hofer’s metrics and boundary depth

Hofer’s metrics and boundary depth

Lagrangian floer Theory over integers: spherically positive symplectic manifolds

Localization of Floer homology of engulfed topological Hamiltonian loop

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