A survey on the Theorem of Chekhanov

TL;DR

This survey reviews Chekhanov's theorem on positive displacement energy of Lagrangian submanifolds, highlighting proofs, Floer homology, and its relation to Morse homology, emphasizing developments in symplectic topology.

Contribution

It provides a comprehensive overview of Chekhanov's theorem, including a direct proof and the development of a filtered Lagrangian Floer homology without assumptions on L.

Findings

Proof based on holomorphic curves and Hamiltonian perturbations

Definition of a filtered Lagrangian Floer homology independent of L's assumptions

Comparison between Floer homology and Morse homology via continuation and PSS maps

Abstract

The theorem of Chekhanov asserts that a Lagrangian submanifold L has positive displacement energy under natural assumptions on the symplectic topology at infinity. It is greater than or equal to the minimal area of holomorphic disks bounded by L. This estimate was obtained by Y.V. Chekhanov in 1998. Section 1 presents a direct proof based on the use of holomorphic curves and their Hamiltonian perturbations. In section 2, we define a filtered version of the Lagrangian Floer homology, without any assumption on L. This is compared with the Morse homology groups, via the continuation maps (subsubsection 2.3.2) or the PSS maps (subsection 3.2).