Lagrangian Flows, Maslov Index Zero and Special Lagrangians

TL;DR

This paper introduces a new notion of vanishing Maslov index for Lagrangian varifolds, constructs mass-decreasing flows leading to special Lagrangians, and connects these flows to homology and conjectures in Lagrangian geometry.

Contribution

It defines vanishing Maslov index for Lagrangian varifolds and cycles, constructs associated mass-decreasing flows, and relates the flow limits to homology and conjectures.

Findings

Flow of cycles converges to special Lagrangian cycles.

Special Lagrangian cycles generate certain Lagrangian homology.

A weak version of the Thomas-Yau conjecture is established.

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.