Computing the Rabinowitz Floer homology of tentacular hyperboloids

TL;DR

This paper calculates Rabinowitz Floer homology for certain non-compact hyperboloids, revealing non-trivial results and confirming the Weinstein Conjecture for their deformations.

Contribution

It introduces a method to compute Floer homology for non-compact hyperboloids and demonstrates the Weinstein Conjecture in this context.

Findings

Rabinowitz Floer homology of hyperboloids is non-zero and differs from singular homology.

A chain map from the Floer complex of hyperboloids to that of spheres is constructed.

The Weinstein Conjecture holds for strongly tentacular deformations of these hyperboloids.

Abstract

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.