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Displacement energy of Lagrangian 3-spheres

Yuhan Sun

Abstract

We estimate the displacement energy of Lagrangian 3-spheres in a symplectic 6-manifold $X$, by estimating the displacement energy of a one-parameter family $L_λ$ of Lagrangian tori near the sphere. The proof establishes a new version of Lagrangian Floer theory with cylinder corrections, which is motivated by the change of open Gromov-Witten invariants under the conifold transition. We also make observations and computations on the classical Floer theory by using symplectic sum formula and Welschinger invariants.

Displacement energy of Lagrangian 3-spheres

Abstract

We estimate the displacement energy of Lagrangian 3-spheres in a symplectic 6-manifold , by estimating the displacement energy of a one-parameter family of Lagrangian tori near the sphere. The proof establishes a new version of Lagrangian Floer theory with cylinder corrections, which is motivated by the change of open Gromov-Witten invariants under the conifold transition. We also make observations and computations on the classical Floer theory by using symplectic sum formula and Welschinger invariants.

Paper Structure

This paper contains 22 sections, 35 theorems, 263 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Future questions
  4. Preliminaries
  5. Computations of classical Floer cohomology
  6. Symplectic cut and sum construction
  7. Welschinger invariants and the pearl complex
  8. A deformed Floer complex
  9. Monotone Lagrangian tori in $T^{*}S^{3}$
  10. Conifold transition
  11. An example about the quadric hypersurface
  12. Weakly unobstructedness of local tori
  13. Holomorphic disks and cylinders
  14. Computations for a local torus
  15. A second deformed Floer complex
  16. ...and 7 more sections

Key Result

Theorem \oldthetheorem

Let $X$ be a closed symplectic 6-manifold with an integrally homologically trivial Lagrangian 3-sphere $S$, and let $L$ be an oriented Lagrangian submanifold of $X$ satisfying Condition condition. For a 4-chain $K$ such that $\partial K=S$ and $K\cap L=\emptyset$, let and $E:=\min\lbrace E_{S}+ v(w), 2v(w), E_{+}\rbrace$. Then there is an $A_{\infty}$-structure $\lbrace \mathfrak{m}^{cy, \mathfra

Figures (13)

  • Figure 1: Disk contributions from inside and outside.
  • Figure 2: Degeneration of a holomorphic disk.
  • Figure 3: Degeneration when circle ends meet.
  • Figure 4: Splittings of disks and cylinders.
  • Figure 5: Composition of parallel transport maps.
  • ...and 8 more figures

Theorems & Definitions (60)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • ...and 50 more