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A new metric on the contactomorphism group of orderable contact manifolds

Lukas Nakamura

TL;DR

The work defines energy-type pseudo-metrics $\widetilde{\delta}^{\pm}_\alpha$ and the induced $\widetilde{d}_\alpha$ on the universal cover, and analogous quantities on ${\rm Cont_0}(M,\xi)$, to quantify nonnegative/positive Hamiltonian dynamics in contact geometry. It proves a dichotomy: on compact $(M,\xi)$, these metrics are either identically zero (iff a contractible positive loop exists) or are $\pi$-non-degenerate (linking to (strong) orderability), with the interval topology agreeing with the metric topology when $M$ is compact; these metrics are bounded above by the Shelukhin–Hofer metric and are closely tied to Hedicke’s Lorentzian distance. The paper extends the framework to orbit spaces of subsets and Legendrian submanifolds, establishing Chekanov-type dichotomies and showing that nondegeneracy corresponds to coisotropic/Legendrian geometric constraints; it also analyzes the Reeb flow as a length-minimizing geodesic and provides bounds for relative growth of contactomorphisms. Overall, the results illuminate the interplay between orderability, interval/Lorentzian topologies, and metric geometry on contactomorphism groups and Legendrian spaces, with implications for the rigidity and dynamics of contactomorphisms. The constructions offer a robust, quantifiable bridge between contact dynamics and geometric topology in both universal covers and base groups.

Abstract

We introduce a pseudo-metric on the contactomorphism group of any contact manifold $(M,ξ)$ with a cooriented contact structure $ξ$. It is the contact analogue of a corresponding semi-norm in Hofer's geometry, and on certain classes of contact manifolds, its lift to the universal cover can be viewed as a continuous version of the integer valued bi-invariant metric introduced by Fraser, Polterovich, and Rosen. We show that it is non-degenerate if and only if $(M,ξ)$ is strongly orderable and that its metric topology agrees with the interval topology introduced by Chernov and Nemirovski. In particular, the interval topology is Hausdorff whenever it is non-trivial, which answers a question of Chernov and Nemirovski. We discuss analogous results for isotopy classes of Legendrians and universal covers.

A new metric on the contactomorphism group of orderable contact manifolds

TL;DR

The work defines energy-type pseudo-metrics and the induced on the universal cover, and analogous quantities on , to quantify nonnegative/positive Hamiltonian dynamics in contact geometry. It proves a dichotomy: on compact , these metrics are either identically zero (iff a contractible positive loop exists) or are -non-degenerate (linking to (strong) orderability), with the interval topology agreeing with the metric topology when is compact; these metrics are bounded above by the Shelukhin–Hofer metric and are closely tied to Hedicke’s Lorentzian distance. The paper extends the framework to orbit spaces of subsets and Legendrian submanifolds, establishing Chekanov-type dichotomies and showing that nondegeneracy corresponds to coisotropic/Legendrian geometric constraints; it also analyzes the Reeb flow as a length-minimizing geodesic and provides bounds for relative growth of contactomorphisms. Overall, the results illuminate the interplay between orderability, interval/Lorentzian topologies, and metric geometry on contactomorphism groups and Legendrian spaces, with implications for the rigidity and dynamics of contactomorphisms. The constructions offer a robust, quantifiable bridge between contact dynamics and geometric topology in both universal covers and base groups.

Abstract

We introduce a pseudo-metric on the contactomorphism group of any contact manifold with a cooriented contact structure . It is the contact analogue of a corresponding semi-norm in Hofer's geometry, and on certain classes of contact manifolds, its lift to the universal cover can be viewed as a continuous version of the integer valued bi-invariant metric introduced by Fraser, Polterovich, and Rosen. We show that it is non-degenerate if and only if is strongly orderable and that its metric topology agrees with the interval topology introduced by Chernov and Nemirovski. In particular, the interval topology is Hausdorff whenever it is non-trivial, which answers a question of Chernov and Nemirovski. We discuss analogous results for isotopy classes of Legendrians and universal covers.
Paper Structure (14 sections, 30 theorems, 112 equations)

This paper contains 14 sections, 30 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Distances on the universal cover ${\rm \widetilde{Cont}_0}(M,\xi)$ of ${\rm Cont_0}(M,\xi)$
  4. Distances on ${\rm Cont_0}(M,\xi)$
  5. Distances on orbit spaces of subsets
  6. Hedicke's Lorentzian distance function
  7. Reeb flow
  8. The relative growth of contactomorphisms
  9. Reparametrizations of Hamiltonians
  10. Proofs of Theorems \ref{['thm:non-deg of d_alpha']} and \ref{['thm:non-degeneracy of d_alpha non-universal']}
  11. Proof of the first part of Theorem \ref{['thm:non-deg of d_alpha']}
  12. Proof of the first part of Theorem \ref{['thm:non-degeneracy of d_alpha non-universal']}
  13. Proof of the second parts of Theorems \ref{['thm:non-deg of d_alpha']} and \ref{['thm:non-degeneracy of d_alpha non-universal']}
  14. Proof of Theorem \ref{["thm:chekanov's dichotomy for d_alpha"]}

Key Result

Theorem 1.3

$d_\alpha$ either vanishes identically or is non-degenerate, and it satisfies $d_\alpha \leq d^\alpha_{SH}$. If $M$ is compact, then the topology on ${\rm Cont_0}(M,\xi)$ induced by $d_\alpha$ agrees with the interval topology $\mathcal{O}_\lprec$, and $d_\alpha$ vanishes identically if and only if

Theorems & Definitions (49)

  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Example 2.9
  • ...and 39 more