Displacement energy of Lagrangian 3-spheres

TL;DR

This paper estimates the displacement energy of Lagrangian 3-spheres in symplectic 6-manifolds by developing a new version of Lagrangian Floer theory with cylinder corrections, connecting to Gromov-Witten invariants.

Contribution

It introduces a novel Lagrangian Floer theory with cylinder corrections motivated by conifold transitions and applies it to displacement energy estimation.

Findings

Displacement energy bounds for Lagrangian 3-spheres.

Development of Floer theory with cylinder corrections.

Computations involving symplectic sum and Welschinger invariants.

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 $L_{\lambda}$ 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.