b-Contact Structures on Tentacular Hyperboloids

This paper connects two different approaches to the analysis of Hamiltonian dynamics on non-compact energy hypersurfaces - $b$-symplectic geometry with its singular symplectic form and Floer techniques for tentacular Hamiltonians. More precisely, we show how to equip tentacular hyperboloids with a $b$-contact structure. We construct a $b^3$-symplectic manifold $(X,Z,{\omega }_b)$, such that each connected component of $X\setminus Z$ is symplectomorphic to the standard symplectic space $(T^{\ast }{\mathbb{R}}^n,{\omega }_0)$. For a tentacular hyperboloid $S\subseteq T^{\ast }{\mathbb{R}}^n$ we look at its copies in $X\setminus Z$ and show that their completion in $(X,Z,{\omega }_b)$ is a smooth hypersurface of $b$-contact type.