Non-density results in high dimensional stable Hamiltonian topology

2024/07/01 by Cardona, Robert, Gironella, Fabio
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2407.01357

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 C³-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 C²-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.

Related

• First steps in stable Hamiltonian topology
• Stability is not open or generic in symplectic four-manifolds
• Non-density of C⁰-stable mappings on non-compact manifolds
• Non-density of stable mappings on non-compact manifolds
• Direct Products for the Hamiltonian Density Property