  We study the deformation of the three-dimensional conformal structures by the
Ricci flow. We drive the evolution equation of Cotton-York tensor and the
L1-norm of it under the Ricci flow. In particular, we investigate the behavior
of the L1-norm of the Cotton-York tensor under the Ricci flow on
three-dimensional simply-connected homogeneous manifolds which admit compact
quotients. For a non-homogeneous case, we also investigate the behavior of the
L1-norm of the Cotton-York tensor for the product metric of the Rosenau
solution for the Ricci flow on a two-sphere and the standard metric of a
circle.
