Bounds for tentacular Hamiltonians

Pasquotto, Federica; Wiśniewska, Jagna

doi:10.1142/s179352531950047x

Cited by 6 publications

(32 citation statements)

References 18 publications

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“…The homology of the chain complex (C F * (H , f ), ∂) is called the Rabinowitz Floer homology of H and is denoted by RF H * (H ). For the Hamiltonians H that we consider in Theorem 1.1, the fact that RF H * (H ) is well defined and independent of the auxiliary data used to construct it is proved in [33,32].…”

Section: Rf H * (Hmentioning

confidence: 96%

“…Throughout this section we will denote by (M , ω) any exact symplectic manifold. In order to prove Theorem 3.1, we need compactness results for homotopies of Hamiltonians and almost complex structures, which are stronger than the ones proved in [33]. To obtain these results, we recall the notion of uniform continuity of (PO), as introduced in [32]: DEFINITION 3.2.…”

Section: Parameter Family Of Hamiltonians In the Affine Space Of Compactly Supported Perturbations Of H Then Rf Hmentioning

confidence: 99%

“…The following proof is an adjustment of the proof of [11,Cor. 3.8] to the setting of strongly tentacular Hamiltonians and is based on the results proven in [33,Prop. 3.3].…”

Section: And If U Is a Solution To The Equationmentioning

confidence: 99%

“…Since (5.5) and (5.8) are both satisfied, we can use [33,Thm. 7.1] and conclude that there exists a function f : (−∞, 0] × S 1 → R, such that r ≥ f and the L 2norm of f is bounded on Ω and the bound depends only on a, b and N .…”

Section: Proposition 54 Consider a Compact Subset N ⊆ σ And A Pair Of Connected Componentsmentioning

confidence: 99%

“…The non-compact case is rather less tractable. In [33,32] the class of noncompact hypersurfaces for which the Rabinowitz Floer homology could be defined was extended, and a rather general invariance result was proved. However no explicit computations were presented.…”

Section: Introductionmentioning

confidence: 99%

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Computing the Rabinowitz Floer homology of tentacular hyperboloids

Fauck¹,

Merry²,

Wiśniewska³

2021

JMD

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<p style='text-indent:20px;'>We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids <inline-formula><tex-math id="M1">\begin{document}$ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $\end{document}</tex-math></inline-formula>. Using an embedding of a compact sphere <inline-formula><tex-math id="M2">\begin{document}$ \Sigma_0\simeq S^{2k-1} $\end{document}</tex-math></inline-formula> into the hypersurface <inline-formula><tex-math id="M3">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula>, we construct a chain map from the Floer complex of <inline-formula><tex-math id="M4">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> to the Floer complex of <inline-formula><tex-math id="M5">\begin{document}$ \Sigma_0 $\end{document}</tex-math></inline-formula>. In contrast to the compact case, the Rabinowitz Floer homology groups of <inline-formula><tex-math id="M6">\begin{document}$ \Sigma $\end{document}</tex-math></inline-formula> are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.</p>

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Section: Rf H * (Hmentioning

confidence: 96%

Section: Parameter Family Of Hamiltonians In the Affine Space Of Compactly Supported Perturbations Of H Then Rf Hmentioning

confidence: 99%

“…The following proof is an adjustment of the proof of [11,Cor. 3.8] to the setting of strongly tentacular Hamiltonians and is based on the results proven in [33,Prop. 3.3].…”

Section: And If U Is a Solution To The Equationmentioning

confidence: 99%

Section: Proposition 54 Consider a Compact Subset N ⊆ σ And A Pair Of Connected Componentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Computing the Rabinowitz Floer homology of tentacular hyperboloids

Fauck¹,

Merry²,

Wiśniewska³

2021

JMD

Self Cite

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The Arnold conjecture for singular symplectic manifolds

Brugués,

Miranda,

Oms

2024

J. Fixed Point Theory Appl.

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In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of $$b^m$$ b m -symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for $$b^{2m}$$ b 2 m -symplectic manifolds depending only on the topology of the manifold. Moreover, for $$b^m$$ b m -symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions.

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b-Contact structures on tentacular hyperboloids

Vogel

Wiśniewska²

2023

Journal of Geometry and Physics

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Bounds for tentacular Hamiltonians

Cited by 6 publications

References 18 publications

Computing the Rabinowitz Floer homology of tentacular hyperboloids

Computing the Rabinowitz Floer homology of tentacular hyperboloids

The Arnold conjecture for singular symplectic manifolds

b-Contact structures on tentacular hyperboloids

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