HAMILTONIAN COLLAPSING OF IRRATIONAL LAGRANGIAN SUBMANIFOLDS WITH SMALL 1ST BETTI NUMBER

Among the main symplectic invariants of a closed Lagrange submanifold L of the cotangent of R(n) is the tubular radius R(L) defined as the smallest tube D(r) x C(n-1) of C(n) congruent-to T*R(n) in which L can be pushed by an Hamiltonian diffeotopy of C(n). We show here, using pseudo-holomorphic techniques, that such a submanifold cannot collapse if the first Betti number of L is smaller than 3 and if the Maslov class of L 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 of L.