  An important and natural question in the analysis of Ricci flow singularity
formation in dimensions four and above is as follows: What are the weakest
conditions that provide control of the norm of the Riemann curvature tensor? In
this short note, we show that on a compact manifold, the trace-free Ricci
tensor is controlled in a precise fashion by the other components of the
irreducible decomposition of the curvature tensor, without any hypotheses on
the initial data.
