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Strict contactomorphisms are scarce

Yong-Geun Oh, Yasha Savelyev

Abstract

The notion of non-projectible contact forms on a given compact manifold $M$ is introduced by the first-named author in [Ohb], the set of which he also shows is a residual subset of the set of (coorientable) contact forms, both in the case with a fixed contact structure and in the case without it. In this paper, we prove that for any non-projectible contact form $λ$ the set, denoted by $\text{\rm Cont}^{\text{\rm st}}(M,λ)$, consisting of strict contactomorphisms of $λ$ is a a countable disjoint union of real lines $\mathbb R$, one for each connected component.

Strict contactomorphisms are scarce

Abstract

The notion of non-projectible contact forms on a given compact manifold is introduced by the first-named author in [Ohb], the set of which he also shows is a residual subset of the set of (coorientable) contact forms, both in the case with a fixed contact structure and in the case without it. In this paper, we prove that for any non-projectible contact form the set, denoted by , consisting of strict contactomorphisms of is a a countable disjoint union of real lines , one for each connected component.

Paper Structure

This paper contains 21 sections, 34 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Strategy of the proof
  4. Floer's $C^\varepsilon$ Banach norms in $C^\infty$ topology
  5. Basic contact Hamiltonian geometry and dynamics
  6. Conformal exponents of contactomorphisms
  7. Hamiltonian calculus of contact vector fields
  8. Volume forms, metric divergence and Lie derivative
  9. Floer's $C^\varepsilon$ norm and the off-shell framework
  10. Chain rule for the derivatives of composition maps
  11. Definition of Floer's $C^\varepsilon$ norms
  12. Definition of $C^{(1,\varepsilon)}$-norms
  13. Boundedness of the first-order linear differential operators
  14. Floer's $C^\varepsilon$ Banach bundles of perturbations
  15. Reduction of the equation of strict contactomorphisms
  16. ...and 6 more sections

Key Result

Theorem 1.1

The set of strict contact pairs, i.e., $(\lambda,\psi)$ satisfying the equation eq:strict-contact is a (Frechet) smooth submanifold of where $\operatorname{Diff}(M, \mu_\lambda)$ is the set of volume-preserving diffeomorphisms of the volume form $\mu_\lambda = \lambda \wedge (d\lambda)^n$.

Theorems & Definitions (64)

  • Theorem 1.1: Theorem \ref{['thm:Phi-submersion']}
  • Example 1.2
  • Lemma 1.3
  • Definition 1.4: Nonprojectible contact form
  • Theorem 1.5: Theorem 1.6 & 1.10 oh:nonprojectible
  • Theorem 1.6
  • Corollary 1.7
  • Definition 1.9
  • Definition 1.10
  • Remark 1.11
  • ...and 54 more