  Using the conformally invariant Cotton tensor, we define a geometric flow,
the "Cotton flow", which is exclusive to three dimensions. This flow tends to
evolve the initial metrics into conformally flat ones, and is somewhat
orthogonal to the Yamabe flow, the latter being a flow within a conformal
class. We define an entropy functional, and study the flow of nine homogeneous
spaces both numerically and analytically. In particular, we show that the
arbitrarily deformed homogeneous 3-sphere flows into the round 3-sphere. Two of
the nine homogeneous geometries, which are degenerated by the Ricci flow, are
left intact by the Cotton flow.
