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$C^0$-flexibility of Legendrian discs in $\mathbb{R}^5$

Maksim Stokić

TL;DR

The paper addresses whether a $C^0$-limit of contact diffeomorphisms preserves Legendrianity of submanifolds. It constructs a compactly supported contact homeomorphism of $\\mathbb{R}^5$ that maps a smoothly embedded Legendrian disc to a smooth nowhere Legendrian disc, exploiting the $h$-principle for loose Legendrians in dimension $\ge 5$ and precise $C^0$ control via a concatenation of small contactomorphisms. The main contribution is a counterexample showing $C^0$-flexibility for Legendrian discs, contrasting with the $C^0$-rigidity results for closed Legendrians, and suggesting a boundary between rigidity and flexibility governed by boundary behavior and loose-chart techniques. The methods combine explicit front-projection constructions, wrinkled/loose Legendrian technology, and isotopy-extension controls that may extend to higher odd dimensions, enhancing understanding of $C^0$-contact and symplectic phenomena.

Abstract

We construct a compactly supported contact homeomorphism of $\mathbb{R}^5$, with the standard contact structure, which maps a Legendrian disc to a smooth nowhere Legendrian disc.

$C^0$-flexibility of Legendrian discs in $\mathbb{R}^5$

TL;DR

The paper addresses whether a -limit of contact diffeomorphisms preserves Legendrianity of submanifolds. It constructs a compactly supported contact homeomorphism of that maps a smoothly embedded Legendrian disc to a smooth nowhere Legendrian disc, exploiting the -principle for loose Legendrians in dimension and precise control via a concatenation of small contactomorphisms. The main contribution is a counterexample showing -flexibility for Legendrian discs, contrasting with the -rigidity results for closed Legendrians, and suggesting a boundary between rigidity and flexibility governed by boundary behavior and loose-chart techniques. The methods combine explicit front-projection constructions, wrinkled/loose Legendrian technology, and isotopy-extension controls that may extend to higher odd dimensions, enhancing understanding of -contact and symplectic phenomena.

Abstract

We construct a compactly supported contact homeomorphism of , with the standard contact structure, which maps a Legendrian disc to a smooth nowhere Legendrian disc.
Paper Structure (10 sections, 7 theorems, 69 equations, 4 figures)

This paper contains 10 sections, 7 theorems, 69 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Proof of Theorem \ref{['TheoremMain']}
  4. Proof of Proposition \ref{['PropZigZagStrips']}
  5. Technical Lemmas
  6. Loose charts on $\Lambda_k$
  7. Appendix
  8. Formal Legendrian embeddings
  9. Wrinkled embeddings and wrinkled Legendrians
  10. Twist markings and loose Legendrians

Key Result

Theorem 1.2

There exists a contact homeomorphism $\Phi$ of $\mathbb{R}^5$ with the standard contact form, which maps smoothly embedded Legendrian disc $\Lambda:[-1,1]^2\rightarrow\mathbb{R}^5$ to a smoothly embedded nowhere Legendrian disc $\Lambda_{\infty}:[-1,1]^2\rightarrow\mathbb{R}^5,\,(t,s)\mapsto(0,0,t,s

Figures (4)

  • Figure 1: Front projection of $\gamma_{m_k}$; red dots represent $\mathrm{ZigZags}(I)$
  • Figure 2: Front projection of $\widetilde{\gamma}_{\tau}$ ($\tau<0$ on the left, and $\tau>0$ on the right)
  • Figure 3: Properties of the function $\lambda(s,\tau)$. A neighbourhood of the zero set of $\lambda$ is coloured in violet.
  • Figure 4: Front projection of $\gamma_{m_k}$ (gray area represents a projection of a loose chart)

Theorems & Definitions (37)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Claim 2.1
  • proof
  • Claim 2.2
  • proof
  • Claim 2.3
  • proof
  • Claim 2.4
  • ...and 27 more