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$C^0$-limits of Legendrians and positive loops

Georgios Dimitroglou Rizell, Michael G. Sullivan

TL;DR

This work addresses how Legendrian submanifolds behave under $C^0$-limits of contactomorphisms and contrasts rigidity with flexibility for non-Legendrians. It proves that if a Legendrian $\Lambda$ is mapped by a $C^0$-limit homeomorphism $\Phi_\infty$ to a smooth submanifold, then $\Phi_\infty(\Lambda)$ is Legendrian; it further shows that non-Legendrians admit positive loops, enabling local deformations that vanish along the evolving submanifold. The authors refine the Chekanov–Hofer–Shelukhin pseudo-norm into a parameterized version and establish its degeneracy on non-Legendrians while maintaining non-degeneracy on Legendrians in suitable settings. Collectively, these results illuminate the delicate balance between rigidity and flexibility in contact topology, with implications for Legendrian behavior under dynamics, stability of invariants, and the global geometry of contactomorphism groups.

Abstract

We show that the image of a Legendrian submanifold under a homeomorphism that is the $C^0$-limit of a sequence of contactomorphisms is again Legendrian, if the image of the submanifold is smooth. In proving this, we show that any non-Legendrian submanifold of a contact manifold admits a positive loop and we provide a parametric refinement of the Rosen--Zhang result on the degeneracy of the Chekanov--Hofer--Shelukhin pseudo-norm for non-Legendrians.

$C^0$-limits of Legendrians and positive loops

TL;DR

This work addresses how Legendrian submanifolds behave under -limits of contactomorphisms and contrasts rigidity with flexibility for non-Legendrians. It proves that if a Legendrian is mapped by a -limit homeomorphism to a smooth submanifold, then is Legendrian; it further shows that non-Legendrians admit positive loops, enabling local deformations that vanish along the evolving submanifold. The authors refine the Chekanov–Hofer–Shelukhin pseudo-norm into a parameterized version and establish its degeneracy on non-Legendrians while maintaining non-degeneracy on Legendrians in suitable settings. Collectively, these results illuminate the delicate balance between rigidity and flexibility in contact topology, with implications for Legendrian behavior under dynamics, stability of invariants, and the global geometry of contactomorphism groups.

Abstract

We show that the image of a Legendrian submanifold under a homeomorphism that is the -limit of a sequence of contactomorphisms is again Legendrian, if the image of the submanifold is smooth. In proving this, we show that any non-Legendrian submanifold of a contact manifold admits a positive loop and we provide a parametric refinement of the Rosen--Zhang result on the degeneracy of the Chekanov--Hofer--Shelukhin pseudo-norm for non-Legendrians.
Paper Structure (12 sections, 14 theorems, 17 equations)

This paper contains 12 sections, 14 theorems, 17 equations.

Table of Contents

  1. Terminology and notation
  2. Statements of results
  3. $C^0$-topology
  4. Positive loops
  5. The Chekanov--Hofer--Shelukhin pseudo-norm
  6. Proofs of results
  7. Basic results for contact Hamiltonians
  8. Proof of Theorem \ref{['thm:ZeroHamiltonian']}
  9. Proof of Corollary \ref{['cor:ParameterizedCHS']}
  10. Proof of Theorem \ref{['thm:NonLegPosLoop']}
  11. Proof of Theorem \ref{['thm:C0']}
  12. Proof of Theorem \ref{['thm:squeezing']}

Key Result

Theorem A

Consider a sequence of contactomorphisms supported in a fixed compact set, and let $\Lambda \subset M$ be a properly embedded Legendrian. If $\Phi_k \xrightarrow{C^0} \Phi_\infty$ where $\Phi_\infty$ is a homeomorphism, and if $\Phi_\infty(\Lambda)$ is smooth, then $\Phi_\infty(\Lambda)$ is also Legendrian.

Theorems & Definitions (25)

  • Theorem A
  • Theorem 2.1
  • Remark 2.2
  • Theorem B
  • Theorem C
  • Remark 2.3
  • Corollary 2.4: Liu:Positive
  • Theorem 2.5: Corollary 5.5 in ChernovNemirovski
  • Corollary 2.6
  • Remark 2.7
  • ...and 15 more