Nondensity results in high-dimensional stable Hamiltonian topology

Abstract

We push forward the study of higher dimensional stable Hamiltonian topology by establishing two nondensity results. First, we prove that stable hypersurfaces are not $C^3$-dense in any isotopy class of embedded hypersurfaces on any ambient symplectic manifold of dimension $2n⩾8$. Our second result is that on any manifold of dimension $2m+1⩾5$, the set of non-degenerate stable Hamiltonian structures is not $C^2$-dense among stable Hamiltonian structures in any given stable homotopy class that satisfies a mild assumption. The latter generalizes a result by Cieliebak and Volkov to arbitrary dimensions.