Rabinowitz Floer Homology for Tentacular Hamiltonians

Wiśniewska, Jagna

doi:10.1093/imrn/rnaa132

Cited by 4 publications

(9 citation statements)

References 16 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: 97%

“…In the case H satisfies (c), this reduces to the original definition presented by Cieliebak and Frauenfelder [11], only specialized to T * R m . Meanwhile for strongly tentacular H , the construction 6 comes from [32].…”

Section: Hamiltoniansmentioning

confidence: 99%

“…Given two distinct components Λ − and Λ + of Crit A H , we denote by M (Λ − , Λ + ) the set of all solutions of (2.5) with finite energy and lim s→±∞ u(s) ∈ Λ ± . We define ev ± : M (Λ − , Λ + ) → Λ ± to be the evaluation maps: By [32,Lem. 8.7] the set of such regular couples is comeagre in H ×J .…”

Section: Complex Structuresmentioning

confidence: 99%

“…Compactness up to breaking implies that ∂ 2 = 0, and a continuation argument tells us that the resulting Rabinowitz Floer homology RF H (H ) is independent of the auxiliary data used to define it. We refer the reader to [11] (when H satisfies (c)) and [32] (when H is strongly tentacular) for details.…”

Section: Gradingmentioning

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: 97%

Section: Hamiltoniansmentioning

confidence: 99%

Section: Complex Structuresmentioning

confidence: 99%

Section: Gradingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Computing the Rabinowitz Floer homology of tentacular hyperboloids

Fauck¹,

Merry²,

Wiśniewska³

2021

JMD

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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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Helium and Hamiltonian delay equations

Frauenfelder¹

2021

Isr. J. Math.

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Rabinowitz Floer Homology for Tentacular Hamiltonians

Cited by 4 publications

References 16 publications

Computing the Rabinowitz Floer homology of tentacular hyperboloids

Computing the Rabinowitz Floer homology of tentacular hyperboloids

The Arnold conjecture for singular symplectic manifolds

Helium and Hamiltonian delay equations

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