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Exact Lagrangians in four-dimensional symplectisations

Georgios Dimitroglou Rizell

Abstract

In this note we provide explicit constructions of exact Lagrangian embeddings of tori and Klein bottles inside the symplectisation of an overtwisted contact three-manifold. Note that any closed exact Lagrangian in the symplectisation is displaceable by a Hamiltonian isotopy. We also use positive loops to exhibit elementary examples of topologically linked Legendrians that are dynamically non-interlinked in the sense of Entov-Polterovich.

Exact Lagrangians in four-dimensional symplectisations

Abstract

In this note we provide explicit constructions of exact Lagrangian embeddings of tori and Klein bottles inside the symplectisation of an overtwisted contact three-manifold. Note that any closed exact Lagrangian in the symplectisation is displaceable by a Hamiltonian isotopy. We also use positive loops to exhibit elementary examples of topologically linked Legendrians that are dynamically non-interlinked in the sense of Entov-Polterovich.
Paper Structure (14 sections, 14 theorems, 48 equations, 3 figures)

This paper contains 14 sections, 14 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. First result: Existence of closed exact Lagrangians
  3. Second Result: Dynamical non-interlinkedness of topologically linked Legendrians
  4. Main trick used in the proofs
  5. Prerequisites
  6. Contact manifolds and symplectisations
  7. Contact Hamiltonians
  8. Constructions of Lagrangian cobordisms
  9. Legendrians in complements of overtwisted discs
  10. Proof of Theorem \ref{['thm:main']}
  11. Exact embeddings of tori
  12. Exact embedding of a Klein bottle
  13. Exact embedding of a torus with vanishing Maslov class
  14. Proof of Theorem \ref{['thm:noninterlinkedness']}

Key Result

Theorem 1.1

For any overtwisted contact three-manifold $(Y^3,\alpha)$, there exist exact Lagrangian embeddings of tori and Klein bottles in its symplectisation $S(Y,\alpha)$. Furthermore, for $\epsilon>0$ sufficiently small and where $\Lambda \subset (Y,\alpha)$ is a Legendrian knot with overtwisted complement.

Figures (3)

  • Figure 1: Left: The front projection of the local model of a Legendrian arc $\eta$ (shown as a dashed blue line) with boundary on a pair of Legendrians $\Lambda_0 \sqcup \Lambda_1$. Right: the front projection of the result of a cusp connected sum $\Lambda_0 \,\sharp\, \Lambda_1$.
  • Figure 2: The front projection of the standard Legendrian unknot contained inside a Darboux ball $(\mathbf{R}^3_{q,p,z},dz-p\,dq)$. The dashed Legendrian arc $\eta$ can be used for performing a connected sum $\Lambda \,\sharp_\eta\,\Lambda_{std}$ that preserves the Legendrian isotopy class of $\Lambda$.
  • Figure 3: Left: The front projection of the zero section $j^10 \subset J^1S^1$. Right: The front projection of a representative of the stabilisation of $(j^10)^{stab}\subset J^1S^1$ of the zero-section contained in the subset $p>0$. Analogous representatives exist also for the stabilisation of the opposite sign.

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.5
  • Corollary 1.6
  • proof
  • Theorem 1.7
  • proof
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • ...and 18 more