Computing the Rabinowitz Floer homology of tentacular hyperboloids

Abstract

We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids \begin{document}$ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $\end{document}. Using an embedding of a compact sphere \begin{document}$ \Sigma_0\simeq S^{2k-1} $\end{document} into the hypersurface \begin{document}$ \Sigma $\end{document}, we construct a chain map from the Floer complex of \begin{document}$ \Sigma $\end{document} to the Floer complex of \begin{document}$ \Sigma_0 $\end{document}. In contrast to the compact case, the Rabinowitz Floer homology groups of \begin{document}$ \Sigma $\end{document} 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.