Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-density results in high dimensional stable Hamiltonian topology

Robert Cardona, Fabio Gironella

TL;DR

The paper extends stable Hamiltonian topology to dimensions $2n+1\ge 5$ by proving two key non-density results: stable hypersurfaces are not $C^3$-dense in any isotopy class inside ambient symplectic manifolds of dimension $\ge 8$, and, within any regular stable homotopy class on manifolds of dimension $2n+1\ge 5$, non-degenerate stable Hamiltonian structures are not $C^2$-dense. The approach combines a robust obstruction built from a normally hyperbolic invariant submanifold embedded in a hypersurface’s characteristic foliation with a semi-local model that enforces degeneracy for perturbations of the Reeb dynamics via first integrals. A robust, non-stabilizable 3D Anosov flow on a homology sphere is constructed and then embedded as NH dynamics in higher dimensions, providing the core obstruction that yields non-density for stable hypersurfaces. A separate construction shows that for regular stable homotopy classes one can force degeneracy under $C^2$-perturbations by creating a local family of normally hyperbolic tori associated with first-integral level sets. Together, these results generalize the prior 3D phenomena of Cieliebak–Volkov to arbitrary dimensions and deepen the understanding of stability and degeneracy in high-dimensional stable Hamiltonian topology.

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.

Non-density results in high dimensional stable Hamiltonian topology

TL;DR

The paper extends stable Hamiltonian topology to dimensions by proving two key non-density results: stable hypersurfaces are not -dense in any isotopy class inside ambient symplectic manifolds of dimension , and, within any regular stable homotopy class on manifolds of dimension , non-degenerate stable Hamiltonian structures are not -dense. The approach combines a robust obstruction built from a normally hyperbolic invariant submanifold embedded in a hypersurface’s characteristic foliation with a semi-local model that enforces degeneracy for perturbations of the Reeb dynamics via first integrals. A robust, non-stabilizable 3D Anosov flow on a homology sphere is constructed and then embedded as NH dynamics in higher dimensions, providing the core obstruction that yields non-density for stable hypersurfaces. A separate construction shows that for regular stable homotopy classes one can force degeneracy under -perturbations by creating a local family of normally hyperbolic tori associated with first-integral level sets. Together, these results generalize the prior 3D phenomena of Cieliebak–Volkov to arbitrary dimensions and deepen the understanding of stability and degeneracy in high-dimensional stable Hamiltonian topology.

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 -dense in any isotopy class of embedded hypersurfaces on any ambient symplectic manifold of dimension . Our second result is that on any manifold of dimension , the set of non-degenerate stable Hamiltonian structures is not -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.
Paper Structure (16 sections, 15 theorems, 34 equations)

This paper contains 16 sections, 15 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stable hypersurfaces and Hamiltonian structures
  4. Inputs from dynamics
  5. Normally hyperbolic invariant manifolds.
  6. Rotation number of circle diffeomorphisms.
  7. Non-density of stable hypersurfaces
  8. A robustly non-stable Anosov flow on a 3D homology sphere
  9. An invariant submanifold with Anosov dynamics
  10. Introducing normal hyperbolicity
  11. Proof of Theorem \ref{['thm:main1']}
  12. Non-density of non-degenerate stable Hamiltonian structures
  13. Regular stable homotopy classes
  14. A degeneracy criterion for Hamiltonian structures
  15. Local construction of transversely hyperbolic invariant tori
  16. ...and 1 more sections

Key Result

Theorem 1

Let $(W,\Omega)$ be a symplectic manifold such that $\dim W\geq 8$. Let $M$ be an embedded hypersurface in $W$. Then $M$ is isotopic to a $C^0$-close hypersurface $\widetilde{M}$ that cannot be $C^3$-approximated by stable hypersurfaces.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • Definition 8
  • Theorem 9: Fe and HPS
  • Remark 10
  • ...and 29 more