Gromov-Floer theory and disjunction energy of compact Lagrangian embeddings
An important tool in symplectic topology is Hofer's norm on the group of Hamiltonian diffeomorphisms on a symplectic manifold, and a non-trivial issue is proving its non-degeneracy. \textit{F. Lalonde} and \textit{D. McDuff} [Ann. Math., II. Ser. 141, No. 2, 349-371 (1995; Zbl 0829.53025)] gave a proof for arbitrary symplectic manifolds. Totally different techniques were used by Polterovich and Chekanov to prove the non-degeneracy for the restricted class of so-called tame symplectic manifolds. These are symplectic manifolds coming with an almost complex structure with the property that the associated metric is a complete Riemannian one, with bounded curvature and injectivity radius bounded away from zero. Using Gromov-Floer theory of perturbed Cauchy-Riemann equations, the present paper gives a new, simpler proof of Chekanov's theorem which is about the positivity of the disjunction energy of compact Lagrangian embeddings in tame symplectic manifolds (and implies the non-degeneracy of Hofer's norm). The author expresses the hope that his approach will also extend to arbitrary symplectic manifolds.
scientific article; zbMATH DE number 1130626