Non-density results in high dimensional stable Hamiltonian topology

TL;DR

This paper demonstrates that in high-dimensional symplectic manifolds, stable hypersurfaces and non-degenerate stable Hamiltonian structures are not dense in their respective classes, extending previous results to arbitrary dimensions.

Contribution

It establishes non-density results for stable hypersurfaces and stable Hamiltonian structures in high dimensions, generalizing prior work to all dimensions above certain thresholds.

Findings

Stable hypersurfaces are not $C^3$-dense in any isotopy class in dimensions ≥8.

Non-degenerate stable Hamiltonian structures are not $C^2$-dense in their classes in dimensions ≥5.

Results extend previous two-dimensional findings to higher dimensions.

Abstract

We push forward the study of higher dimensional stable Hamiltonian topology by establishing two non-density 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 $2n\geq 8$. Our second result is that on any manifold of dimension $2m+1\geq 5$, 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.