Lagrangian Flows, Maslov Index Zero and Special Lagrangians

Abstract

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.

v2: Extended Corollary 0.5 from $n=2$ to $n\leq 6$, namely: The lagrangian homology of any simply-connected closed Calabi-Yau manifold is generated by special lagrangian cycles (possibly with different phases), in dimensions $n\leq 6$ v3: Clarified the class of varifolds we use. Added details to some arguments in sections 3 and 5