Computing the Rabinowitz Floer homology of tentacular hyperboloids

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