TITLE: Evolution of $W_+$ and $W_-$ under the Ricci flow
QUESTION [5 upvotes]: In dimension $4$ the Weyl operator $W$ splits in two parts
$$W_+:\Lambda^{2}_{+} \to \Lambda^{2}_{+}$$
and
$$W_-:\Lambda^{2}_{-} \to \Lambda^{2}_{-}.$$
(a) Has there been a study of the evolution under the Ricci flow of these two endomorphisms?
(b) If so, is there a source where these are written explicitly?

REPLY [4 votes]: You can find a computation for the evolution of $A=\tfrac{S}{6}I_++W_+$ in Hamilton's "Four-manifolds with positive curvature operator", it is right before Lemma 3.1. What is found is :
$\partial_tA=\Delta A + A^2+2A^\#+ BB^T$
Where $A^\#$ is the cofactor matrix of the $3\times 3$ matrix of $A$ in an orthonormal frame and $B$ is the matrix of $\tfrac{1}{2}Ric_0\wedge id:\Lambda^2_-\to\Lambda^2_+$.
This can be proved just by "projecting" the evolution of the full curvature tensor on the $S^2(\Lambda^2_+)$ component.
To get the evolution of $W_+$ alone, just subtract to it the evolution of $S$.
Remark : All computations are done assuming Uhlenbeck's trick.