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$C^1$-Local Flatness and Geodesics of the Legendrian Spectral Distance

Simon Allais, Pierre-Alexandre Arlove

Abstract

In this article, we give an explicit computation of the order spectral selectors of a pair of $C^1$-close Legendrian submanifolds belonging to an orderable isotopy class. The $C^1$-local flatness of the spectral distance and the characterisation of its geodesics are deduced. Another consequence is the $C^1$-local coincidence of spectral and Shelukhin-Chekanov-Hofer distances. Similar statements are then deduced for several contactomorphism groups.

$C^1$-Local Flatness and Geodesics of the Legendrian Spectral Distance

Abstract

In this article, we give an explicit computation of the order spectral selectors of a pair of -close Legendrian submanifolds belonging to an orderable isotopy class. The -local flatness of the spectral distance and the characterisation of its geodesics are deduced. Another consequence is the -local coincidence of spectral and Shelukhin-Chekanov-Hofer distances. Similar statements are then deduced for several contactomorphism groups.
Paper Structure (13 sections, 15 theorems, 24 equations)

This paper contains 13 sections, 15 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. $C^1$-flatness
  3. Geodesics
  4. Discussion
  5. Preliminaries
  6. Conventions
  7. Hofer type distances
  8. Order spectral selectors and induced distances
  9. Proof of Theorem \ref{['thm']}
  10. Characterization of the geodesics
  11. The case of contactomorphisms
  12. $C^1$-local flatness
  13. Geodesics

Key Result

Theorem 1.1

If $\mathcal{L}$ (resp. $\widetilde{\mathcal{L}}$) is orderable then endowed with the Legendrian spectral distance it is $C^1$-locally flat. More precisely, for every $\Lambda\in\mathcal{L}$ (resp. $\widetilde{\mathcal{L}}$), and every $\alpha$-Weinstein parametrization $\Phi:U\to\mathcal{L}$ (resp. in particular $\mathrm{d}_\mathrm{spec}^\alpha(\Phi(f),\Phi(g))=\max |f-g|$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • proof
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 2.1
  • ...and 23 more