More on Cotton Flow

TL;DR

This paper analyzes the stability and properties of Cotton flow on three-manifolds, revealing instabilities in flat and certain Einstein spaces, and exploring soliton solutions including topologically massive and pp-wave metrics.

Contribution

It extends previous work on Cotton flow by studying linear stability, refining the gradient flow formalism, and identifying new soliton solutions in both Riemannian and Lorentzian signatures.

Findings

Flat space is linearly unstable under Cotton flow.

Certain Einstein and conformally flat non-Einstein spaces are unstable.

New soliton solutions, including topologically massive and pp-wave metrics, are identified.

Abstract

Cotton flow tends to evolve a given initial metric on a three manifold to a conformally flat one. Here we expound upon the earlier work on Cotton flow and study the linearized version of it around a generic initial metric by employing a modified form of the DeTurck trick. We show that the flow around the flat space, as a critical point, reduces to an anisotropic generalization of linearized KdV equation with complex dispersion relations one of which is an unstable mode, rendering the flat space unstable under small perturbations. We also show that Einstein spaces and some conformally flat non-Einstein spaces are linearly unstable. We refine the gradient flow formalism and compute the second variation of the entropy and show that generic critical points are extended Cotton solitons. We study some properties of these solutions and find a Topologically Massive soliton that is built from Cotton and Ricci solitons. In the Lorentzian signature, we also show that the pp-wave metrics are both Cotton and Ricci solitons.