Tightness of Chekanov's bound on displacement energy for some Lagrangian knots

2025/07/28 by Yaar, David Keren
#53D12 (Primary) 53D20 (Secondary) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2507.21282

Abstract

By a classical theorem of Chekanov, the displacement energy, e, of a Lagrangian submanifold is bounded from below by the minimal area of pseudo-holomorphic disks with boundary on the Lagrangian, ℏ. We compute e and ℏ for displaceable Chekanov tori in ℂPⁿ, and for an infinite family of exotic tori in ℂ³ constructed by Brendel. In these families, e=ℏ. We compare continuity properties of e and ℏ on the space of Lagrangians. This provides an example (suggested by Fukaya, Oh, Ohta, and Ono) where e>ℏ. Our calculations have further applications such as a new proof, inspired by work of Auroux, that Brendel's family of exotic tori consists of infinitely many distinct Lagrangians.

Citations

• Local exotic tori
• Real Lagrangian Tori and Versal Deformations
• Geometry of symplectic flux and Lagrangian torus fibrations
• Displacement of polydisks and Lagrangian Floer theory
• Notes on monotone Lagrangian twist tori
• Special Lagrangian fibrations, wall-crossing, and mirror symmetry
• Lagrangian Floer theory on compact toric manifolds II : Bulk deformations
• Holomorphic disc, spin structures and Floer cohomology of the Clifford torus

Related

• A survey on the Theorem of Chekhanov
• Symplectic displacement energy for exact Lagrangian immersions
• Displacement energy of compact Lagrangian submanifold from open subset
• Local exotic tori
• Energy functional for Lagrangian tori in ℂP²