A survey on the Theorem of Chekhanov

Benoit Tonnelier (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

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).

Mots clés: Lagrangian submanifold; displacement energy; Lagrangian Floer homology; holomorphic curves

Domaines: Géométrie symplectique [math.SG]

Benoit Tonnelier. A survey on the Theorem of Chekhanov. 2009. ⟨hal-00437937⟩