Symplectic topology of Mañé’s critical values

Cieliebak, Kai; Frauenfelder, Urs; Paternain, Gabriel P.

doi:10.2140/gt.2010.14.1765

Cited by 69 publications

(153 citation statements)

References 71 publications

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“…This is not useful here, because the a priori estimates which lead to the existence of the pseudo-gradient vector field W come from a contradiction argument. However, it might be useful in situations where these a priori bounds have a different origin, such as for example in the case of tame energy levels (see [CFP10] for the definition of tameness and for motivating examples).…”

Section: Periodic Orbits With Low Energymentioning

confidence: 99%

Lectures on the free period Lagrangian action functional

Abbondandolo

2013

J. Fixed Point Theory Appl.

107

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Section: Periodic Orbits With Low Energymentioning

confidence: 99%

Lectures on the free period Lagrangian action functional

Abbondandolo

2013

J. Fixed Point Theory Appl.

107

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“…For example, if the universal cover of L is contractible the inclusion i : L → Λ c L induces an isomorphism i * : H * (L) → H * (Λ 0 L). On the other hand, H 0 (Λ c L), and hence also RF H c * (DT * L), is nonzero for each nontrivial free homotopy class c. Corollary 1.12 is used in [11] to study the dynamics of magnetic flows. In order to apply it to exact contact embeddings, we need the a criterion for independence of Rabinowitz Floer homology of the symplectic filling V given in the following result.…”

Section: Introductionmentioning

confidence: 99%

Rabinowitz Floer homology and symplectic homology

Cieliebak¹,

Frauenfelder²,

Oancea³

2010

Ann. Sci. École Norm. Sup.

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179

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Abstract. The first two authors have recently defined RabinowitzFloer homology groups RF H * (M, W ) associated to an exact embedding of a contact manifold (M, ξ) into a symplectic manifold (W, ω). These depend only on the bounded component V of W \ M . We construct a long exact sequence in which symplectic cohomology of V maps to symplectic homology of V , which in turn maps to Rabinowitz-Floer homology RF H * (M, W ), which then maps to symplectic cohomology of V . We compute RF H * (ST * L, T * L), where ST * L is the unit cosphere bundle of a closed manifold L. As an application, we prove that the image of an exact contact embedding of ST * L (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided dim L ≥ 4 and the embedding induces an injection on π1. In particular, ST * L does not admit an exact contact embedding into a subcritical Stein manifold if L is simply connected. We also prove that Weinstein's conjecture holds in symplectic manifolds which admit exact displaceable codimension 0 embeddings. IntroductionLet (W, λ) be a complete convex exact symplectic manifold, with symplectic form ω = dλ (see Section 3 for the precise definition). An embedding ι : M ֒→ W of a contact manifold (M, ξ) is called exact contact embedding if there exists a 1-form α on M such that such that ker α = ξ and α − λ| M is exact. We identify M with its image ι(M ). We assume that W \ M consists of two connected components and denote the bounded component of W \ M by V . One can classically [25] associate to such an exact contact embedding the symplectic (co)homology groups SH * (V ) and SH * (V ). We The first two authors have recently defined for such an exact contact embedding Floer homology groups RF H * (M, W ) for the Rabinowitz action functional [9]. We refer to Section 3 for a recap of the definition and of some useful properties. We will show in particular that these groups do not depend on W , but only on V (the same holds for SH * (V ) and SH * (V )). We shall use in this paper the notation RF H * (V ) and call them Rabinowitz Floer homology groups. Remark 1.1. All (co)homology groups are taken with field coefficients. Without any further hypotheses on the first Chern class c 1 (V ) of the tangent bundle, the symplectic (co)homology and Rabinowitz Floer homology groups are Z 2 -graded. If c 1 (V ) = 0 they are Z-graded, and if c 1 (V ) vanishes on π 2 (V ) the part constructed from contractible loops is Z-graded. This Zgrading on Rabinowitz Floer homology differs from the one in [9] (which takes values in 1 2 + Z) by a shift of 1/2 (see Remark 3.2).Our purpose is to relate these two constructions. The relevant object is a new version of symplectic homology, denotedŠH * (V ), associated to " -shaped" Hamiltonians like the one in Figure 1 on page 20 below. This version of symplectic homology is related to the usual ones via the long exact sequence in the next theorem. Theorem 1.2. There is a long exact sequence (1). . .The long exact sequence (1) can be seen ...

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“…If M has dimension higher than 2 and σ is a symplectic form, proving that low energy levels are stable is an open problem (see [CFP10] where a proof of stability is given in the homogeneous case). However, existence of contractible magnetic geodesics for every low energy levels still holds as Usher proves in [Ush09] building upon previous work of Ginzburg and Gürel [GG09] (see also [Ker99] for multiplicity results, generalizing [Gin87], when σ is a Kähler form).…”

Section: H(q P)mentioning

confidence: 99%

The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

Asselle

Benedetti

2016

J. Topol. Anal.

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Abstract. Let M be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian H : T * M → R and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits for such a flow. Our main result asserts that if M is not aspherical then contractible periodic orbits exist for almost all energies above the maximum critical value of H.

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Symplectic topology of Mañé’s critical values

Cited by 69 publications

References 71 publications

Lectures on the free period Lagrangian action functional

Lectures on the free period Lagrangian action functional

Rabinowitz Floer homology and symplectic homology

The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles

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