  We introduce a notion of vanishing Maslov index for lagrangian varifolds and
lagrangian integral cycles in a Calabi-Yau manifold. We construct
mass-decreasing flows of lagrangian varifolds and lagrangian cycles which
satisfy this condition. The flow of cycles converges, at infinite time, to a
sum of special lagrangian cycles (possibly with differing phases).
  We use the flow of cycles to obtain the fact that special lagrangian cycles
generate the part of the lagrangian homology which lies in the image of the
Hurewicz homomorphism. We also establish a weak version of a conjecture of
Thomas-Yau regarding lagrangian mean curvature flow.
