Hamiltonian collapsing of irrational Lagrangian submanifolds with small first Betti number

Abstract

Among the main symplectic invariants of a closed Lagrange submanifoldL of the cotangent of ℝn is the tubular radiusR(L) defined as the smallest tube D(r) × ℂn−1 of ℂn ≃ T* ℝn in whichL can be pushed by an Hamiltonian diffeotopy of ℂn. We show here, using pseudo-holomorphic techniques, that such a submanifold cannot collapse if the first Betti number ofL is smaller than 3 and if the Maslov class ofL does not vanish; in other words,R(L) is then strictly positive and one can actually give an explicit lower bound in terms of the Liouville and Maslov classes ofL.