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

Lagrangian Flows, Maslov Index Zero and Special Lagrangians