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Non-orderability and the contact Hofer norm

Jakob Hedicke, Egor Shelukhin

TL;DR

This work establishes a deep link between non-orderability in contact topology and shortening phenomena in the contact Hofer norm. By leveraging open book decompositions, the loose Legendrian $h$-principle, and fragmentation, the authors derive broad non-orderability results for ideal boundaries of subcritical Weinstein manifolds, including the standard $S^1\times S^2$, and for prequantizations with subcritical polarizations. They also demonstrate a robust $\mathcal{C}^0$-continuity property of the contact Hofer metric and discuss explicit constructions of positive loops, translated-point obstructions, and Weinstein-conjecture-type consequences. Collectively, the results expand the catalog of non-orderable contact manifolds and reveal structural links between dynamics of Reeb flows, skeletal geometry of pages, and rigidity/flexibility phenomena in contact topology. The findings have implications for symplectic fillability, pre-Lagrangian displacement, and the geometry of contactomorphism groups, with potential to inform future inquiries into loose versus flexible phenomena and their impact on orderability and polarizations.

Abstract

We relate non-orderability in contact topology to shortening in the contact Hofer norm. Combined with considerations of open books, this provides many new examples of non-orderable contact manifolds, including contact boundaries of subcritical Weinstein domains, and in particular the long-standing case of the standard $S^1 \times S^2.$ We also produce new examples of contact manifolds admitting contactomorphisms without translated points, provide obstructions to subcritical polarizations of symplectic manifolds, and establish a $\mathcal{C}^0$-continuity property of the contact Hofer metric.

Non-orderability and the contact Hofer norm

TL;DR

This work establishes a deep link between non-orderability in contact topology and shortening phenomena in the contact Hofer norm. By leveraging open book decompositions, the loose Legendrian -principle, and fragmentation, the authors derive broad non-orderability results for ideal boundaries of subcritical Weinstein manifolds, including the standard , and for prequantizations with subcritical polarizations. They also demonstrate a robust -continuity property of the contact Hofer metric and discuss explicit constructions of positive loops, translated-point obstructions, and Weinstein-conjecture-type consequences. Collectively, the results expand the catalog of non-orderable contact manifolds and reveal structural links between dynamics of Reeb flows, skeletal geometry of pages, and rigidity/flexibility phenomena in contact topology. The findings have implications for symplectic fillability, pre-Lagrangian displacement, and the geometry of contactomorphism groups, with potential to inform future inquiries into loose versus flexible phenomena and their impact on orderability and polarizations.

Abstract

We relate non-orderability in contact topology to shortening in the contact Hofer norm. Combined with considerations of open books, this provides many new examples of non-orderable contact manifolds, including contact boundaries of subcritical Weinstein domains, and in particular the long-standing case of the standard We also produce new examples of contact manifolds admitting contactomorphisms without translated points, provide obstructions to subcritical polarizations of symplectic manifolds, and establish a -continuity property of the contact Hofer metric.

Paper Structure

This paper contains 28 sections, 27 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the proofs
  3. Future directions
  4. Background
  5. Orderability
  6. The contact Hofer norm
  7. Liouville and Weinstein manifolds
  8. Open book decompositions and contact structures
  9. Loose Legendrians
  10. Bounds on the contact Hofer norm
  11. The contact Hofer diameter of skeleton-complements
  12. A Fragmentation result
  13. Proof of Theorem \ref{['thm1']} (i)
  14. Proof of Theorem \ref{['thm1']} (ii)
  15. Proof of Corollary \ref{['cor1stab']}
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $(M^{2n+1},\xi)$ be a closed contact manifold, $W$ a page of a Weinstein open book supporting $\xi$ and $L$ the skeleton of $W$ with respect to some ideal Giroux form. Then the following holds. In both cases $(M,\xi)$ is non-orderable.

Figures (1)

  • Figure 1: Front projection of the Legendrian arc $\gamma$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 35 more