Let $M$ be a smooth $n$-dimensional manifold, let $I \subset \mathbb{R}$ be an interval, and let

\begin{align*} g: I \to \Gamma(S^2T^*M) \end{align*}

be a smooth solution of the Ricci flow

\begin{align*} \partial_t g(t) = -2\operatorname{Ric}(g(t)). \end{align*}

For each $t \in I$, let $\operatorname{Rm}$ denote the Riemann curvature tensor of $g(t)$, let $\nabla$ denote the Levi-Civita connection of $g(t)$, and let $\Delta = \operatorname{tr}_{g(t)} \nabla\nabla$ denote the rough Laplacian acting on tensor fields. Then there exists a constant $C_n > 0$, depending only on $n$, such that at every point $(p,t) \in M \times I$,

\begin{align*} (\partial_t - \Delta)|\nabla \operatorname{Rm}|^2 \leq -2|\nabla\nabla \operatorname{Rm}|^2 + C_n|\operatorname{Rm}|\,|\nabla \operatorname{Rm}|^2, \end{align*}

where all norms and contractions are taken with respect to $g(t)$.