Cotton flow

TL;DR

The paper introduces the Cotton flow, a geometric evolution in three dimensions that drives metrics toward conformally flat states, and analyzes its behavior on homogeneous spaces both numerically and analytically.

Contribution

It defines the Cotton flow using the conformally invariant Cotton tensor and studies its effects on homogeneous spaces, including convergence to round spheres.

Findings

The flow evolves deformed 3-spheres into round 3-spheres.

Some degenerated geometries remain unchanged under the Cotton flow.

An entropy functional for the flow is established.

Abstract

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.