This paper represents a first step towards the extension of the definition of Rabinowitz Floer homology to non-compact energy hypersurfaces Σ in exact symplectic manifolds. More concretely, we study under which conditions it is possible to establish L ∞ -bounds for the Floer trajectories of a Hamiltonian with non-compact energy levels. Moreover, we introduce a class of Hamiltonians, called tentacular Hamiltonians which satisfy the conditions: how to define RFH for these examples will be the subject of a follow-up paper.

This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces arising as the level sets of specific Hamiltonians: strongly tentacular Hamiltonians for which the compactness conditions are satisfied, cf. [ 21], thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a tool to address the Weinstein conjecture and establish existence of closed characteristics for non-compact contact manifolds.

<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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We compute the Rabinowitz Floer homology for a class of noncompact hyperboloids Σ ≃ S n+k−1 × R n−k . Using an embedding of a compact sphere Σ0 ≃ S 2k−1 into the hypersurface Σ, we construct a chain map from the Floer complex of Σ to the Floer complex of Σ0. In contrast to the compact case, the Rabinowitz Floer homology groups of Σ 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.