Loops of Lagrangian submanifolds and pseudoholomorphic discs

Details Loops of Lagrangian submanifolds and pseudoholomorphic discs Loops of Lagrangian submanifolds and pseudoholomorphic discs

2000-03-14

arxiv.org/abs/math/0003079v2

Mathematics

Symplectic Geometry

51 pages, one LaTeX file and two pictures

Scientific paper

The main theorem of this paper asserts that the inclusion of the space of projective Lagrangian planes into the space of Lagrangian submanifolds of complex projective space induces an injective homomorphism of fundamental groups. We introduce 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.

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