Symplectic Displacement Energy for Lagrangian Submanifolds

摘要

Recently H. Hofer introduced an interesting quantity, so called displacementenergy, associated with a subset of a symplectic manifold. Roughly speaking, the quantity measures how large a variation of a compactly supported Hamiltonian function must be in order to push the subset away from itself by a time-one map of the corresponding Hamiltonian flow. Hofer proved that, at least in the standard symplectic vector space, displacement energy of every open subset is strictly positive. In the present paper the authors study displacement energy for Lagrangian submanifolds of symplectic manifolds. The authors estimate it from below through symplectic area class assuming that the symplectic manifold is tame and the Lagrangian submanifold is closed and rational. As a corollary the authors obtain that if symplectic area class of a tame symplectic manifold is rational, then the displacement energy of every open subset is positive.