The Cotton tensor and the Ricci flow

TL;DR

This paper derives how the Cotton and Bach tensors evolve under Ricci flow on Riemannian manifolds, focusing on three dimensions, and explores relevant applications of these evolution equations.

Contribution

It provides explicit evolution equations for the Cotton and Bach tensors under Ricci flow, especially in three dimensions, which was not previously detailed.

Findings

Derived evolution equations for Cotton and Bach tensors

Analyzed the implications in three-dimensional Ricci flow

Discussed applications of these evolution equations

Abstract

We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.