  In this paper, we construct various examples of Lagrangian mean curvature
flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie
groups on them. The examples include Lagrangian self-shrinkers and translating
solitons in the Euclidean spaces. Moreover, our method can be applied to
construct examples of Lagrangian mean curvature flows in non-flat Calabi-Yau
manifolds. In particular, we describe Lagrangian mean curvature flows in
4-dimensional Ricci-flat ALE spaces in detail and investigate their
singularities.
