Hamiltonian Isotopy Research Articles

Loops of Lagrangian submanifolds and pseudoholomorphic discs

We examine three invariants of exact loops of Lagrangian submanifolds that are modelled on invariants introduced by Polterovich for loops of Hamiltonian symplectomorphisms. One of these is the minimal Hofer length in a given Hamiltonian isotopy class. We determine the exact values of these invariants for loops of projective Lagrangian planes. The proof uses the Gromov invariants of an associated symplectic fibration over the 2-disc with a Lagrangian subbundle over the boundary.

A Lagrangian camel

We prove the Lagrangian analogue of the symplectic camel theorem: there are compact Lagrangian submanifolds of $ {\Bbb R}^{2n} $ that cannot be moved through a small hole by a global Hamiltonian isotopy with compact support.

Hamiltonian isotopies in [formula omitted]2 n+: Negative results

We present a failure in a problem of extension of Hamiltonian isotopies in a framework where positive answer could reasonably be expected. Symplectic differential topology remains well hidden.