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Liouville polarizations and the rigidity of their Lagrangian skeleta in dimension $4$

Emmanuel Opshtein, Felix Schlenk

Abstract

The main theme of this paper is the introduction of a new type of polarizations, suited for some open symplectic manifolds, and their applications. These applications include symplectic embedding results that answer a question by Sackel-Song-Varolgunes-Zhu and Brendel, new Lagrangian non-removable intersections at small scales, and a novel phenomenon of Legendrian barriers in contact geometry.

Liouville polarizations and the rigidity of their Lagrangian skeleta in dimension $4$

Abstract

The main theme of this paper is the introduction of a new type of polarizations, suited for some open symplectic manifolds, and their applications. These applications include symplectic embedding results that answer a question by Sackel-Song-Varolgunes-Zhu and Brendel, new Lagrangian non-removable intersections at small scales, and a novel phenomenon of Legendrian barriers in contact geometry.
Paper Structure (25 sections, 32 theorems, 139 equations, 13 figures)

This paper contains 25 sections, 32 theorems, 139 equations, 13 figures.

Table of Contents

  1. Introduction and main results
  2. Symplectic embeddings
  3. Rigidity properties of Lagrangian skeleta
  4. Legendrian barriers
  5. A general embedding result.
  6. Related results and references
  7. Polarisations of symplectic manifolds
  8. Neighborhoods of symplectic curves in dimension $4$
  9. Polarisations (closed case)
  10. Liouville polarisations (open case)
  11. Some explicit polarisations and skeleta
  12. Polarisations in dimension $2$
  13. Bidiscs
  14. Surgery on Liouville polarisations
  15. Symplectic embeddings
  16. ...and 10 more sections

Key Result

Theorem 1

There exists an $(\alpha_{{\operatorname{st}}}, \alpha_{{\operatorname{st}}})$-exact symplectic embedding

Figures (13)

  • Figure 1.1: Examples of grids.
  • Figure 2.1: An extension $(\widehat{\Omega}, \widehat{\Sigma})$ of $(\Omega, \Sigma)$.
  • Figure 2.2: The symplectomorphism $f \colon V \to V'$ with $f^* (\lambda + \vartheta) = \lambda$.
  • Figure 2.3: The extension $\psi \colon U_0 \to U_1$ of $\varphi \colon \widehat{X}_0 \to \widehat{X}_1$.
  • Figure 2.4: ${\textcolor{red}{X}} \subset \widehat{\Sigma}$ and $\operatorname{supp} \vartheta$.
  • ...and 8 more figures

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1.3
  • Theorem 4
  • Example
  • Theorem 5
  • Definition
  • Theorem 6
  • Example 1.5
  • ...and 38 more