I think under Ricci flow ,the rank of curvature operator does not change by +1 or -1, it will directly change to full rank or zero rank . I want to write it as term paper, but I don't know whether there are somebody have proved it or it is very trivial or hard. Besides, I just read a little of Cao and Zhu's A COMPLETE PROOF OF THE ... , what paper or reference maybe useful for this question ?
The rank of curvature is the rank of $M_{\alpha\beta}$ in picture below.Picture below is from the 185 and 213 of this paper.
