[proofplan]
We first record the tensor evolution equation for $\nabla \operatorname{Rm}$ in schematic form, keeping explicit control of the dimensional constants hidden in the contraction notation. The Ricci flow curvature equation gives $(\partial_t-\Delta)\operatorname{Rm}=\operatorname{Rm}*\operatorname{Rm}$; after covariantly differentiating and commuting $\nabla$ through $\partial_t-\Delta$, all terms become contractions of one factor $\operatorname{Rm}$ and one factor $\nabla\operatorname{Rm}$. We then apply the time-dependent Bochner identity to $|\nabla\operatorname{Rm}|^2$, which produces the coercive term $-2|\nabla\nabla\operatorname{Rm}|^2$. Finally, the remaining contraction terms are estimated pointwise by a constant depending only on the dimension.
[/proofplan]
[step:Introduce controlled contraction notation]
For tensor fields $A$ and $B$ on $(M,g(t))$, write $A*B$ for a finite linear combination of tensors obtained from $A \otimes B$ by contracting indices using $g(t)$ and $g(t)^{-1}$, with coefficients depending only on $n$ and on the tensor types involved. Thus, whenever a tensor $E$ satisfies $E=A*B$, there is a constant $c=c(n)>0$ such that
\begin{align*}
|E| \leq c |A| |B|.
\end{align*}
Likewise, $A*B*C$ denotes a finite linear combination of contractions of $A\otimes B\otimes C$, and therefore satisfies
\begin{align*}
|A*B*C| \leq c |A| |B| |C|
\end{align*}
for a dimensional constant $c=c(n)>0$.
In the rest of the proof, every occurrence of $*$ is used in this controlled sense.
[/step]
[step:Commute the covariant derivative through the heat operator]
By the Ricci flow curvature evolution identity, the curvature tensor satisfies
\begin{align*}
(\partial_t-\Delta)\operatorname{Rm}
=
\operatorname{Rm}*\operatorname{Rm}.
\end{align*}
Here the identity is applied to the smooth time-dependent metric $g(t)$ along the Ricci flow, and the right-hand side has the tensor type of the Riemann curvature tensor.
Covariantly differentiating gives
\begin{align*}
\nabla\bigl((\partial_t-\Delta)\operatorname{Rm}\bigr)
=
\nabla\operatorname{Rm}*\operatorname{Rm}
+
\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
Since both terms on the right are controlled contractions of $\operatorname{Rm}$ and $\nabla\operatorname{Rm}$, this is
\begin{align*}
\nabla\bigl((\partial_t-\Delta)\operatorname{Rm}\bigr)
=
\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
It remains to account for the commutator between $\nabla$ and $\partial_t-\Delta$. Let $(x_1,\dots,x_n)$ be local coordinates on an open coordinate neighbourhood $U \subset M$, and let $\Gamma_{ij}^{k}: U \times I \to \mathbb{R}$ denote the Christoffel symbols of the Levi-Civita connection of $g(t)$ in these coordinates. By the variation formula for the Levi-Civita connection under Ricci flow, these symbols satisfy the schematic identity
\begin{align*}
\partial_t \Gamma = \nabla \operatorname{Ric},
\end{align*}
meaning each component $\partial_t\Gamma_{ij}^{k}$ is a universal linear combination of components of $\nabla\operatorname{Ric}$ with indices raised using $g(t)^{-1}$. Hence, for every time-dependent tensor field $T$, the induced commutator has the controlled schematic form
\begin{align*}
[\partial_t,\nabla]T = \nabla\operatorname{Ric}*T.
\end{align*}
The Ricci tensor is a contraction of $\operatorname{Rm}$, so $\nabla\operatorname{Ric}$ is a contraction of $\nabla\operatorname{Rm}$. Therefore, with $T=\operatorname{Rm}$,
\begin{align*}
[\partial_t,\nabla]\operatorname{Rm}
=
\nabla\operatorname{Rm}*\operatorname{Rm}.
\end{align*}
By the connection commutator formula for the rough Laplacian, applied to the covariant derivative of the curvature tensor,
\begin{align*}
[\Delta,\nabla]\operatorname{Rm}
=
\operatorname{Rm}*\nabla\operatorname{Rm}
+
\nabla\operatorname{Rm}*\operatorname{Rm}
=
\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
Combining these identities, the definition of the commutator gives
\begin{align*}
(\partial_t-\Delta)\nabla\operatorname{Rm}
=
\nabla\bigl((\partial_t-\Delta)\operatorname{Rm}\bigr)
+
[\partial_t-\Delta,\nabla]\operatorname{Rm}.
\end{align*}
Substituting the two schematic identities already obtained yields
\begin{align*}
(\partial_t-\Delta)\nabla\operatorname{Rm}
=
\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
[/step]
[step:Apply the time-dependent Bochner identity to $|\nabla\operatorname{Rm}|^2$]
Let $\pi: M \times I \to M$ denote the projection map $(p,t) \mapsto p$, and let
\begin{align*}
A \in \Gamma\left((\pi^*T^*M)^{\otimes 5}\right)
\end{align*}
be the time-dependent covariant tensor field defined by
\begin{align*}
A = \nabla\operatorname{Rm}.
\end{align*}
By the tensor Bochner identity, equivalently the product rule for the rough Laplacian applied to the pointwise norm of a tensor field,
\begin{align*}
\Delta |A|^2
=
2(\Delta A,A)
+
2|\nabla A|^2,
\end{align*}
where $(\cdot,\cdot)$ denotes the pointwise [inner product](/page/Inner%20Product) induced by $g(t)$. The time derivative of the norm also differentiates the metric contractions. Since $\partial_t g=-2\operatorname{Ric}$ and $\partial_t g^{-1}=2\operatorname{Ric}^{\sharp}$, where $\operatorname{Ric}^{\sharp}$ denotes the $(2,0)$-tensor obtained from the Ricci tensor by raising both indices with $g(t)^{-1}$, the metric-variation contribution is a controlled contraction
\begin{align*}
\operatorname{Ric}*A*A
=
\operatorname{Rm}*A*A.
\end{align*}
Therefore,
\begin{align*}
(\partial_t-\Delta)|A|^2
=
2\bigl((\partial_t-\Delta)A,A\bigr)
-
2|\nabla A|^2
+
\operatorname{Rm}*A*A.
\end{align*}
Substituting $A=\nabla\operatorname{Rm}$ yields
\begin{align*}
(\partial_t-\Delta)|\nabla\operatorname{Rm}|^2
=
2\bigl((\partial_t-\Delta)\nabla\operatorname{Rm},\nabla\operatorname{Rm}\bigr)
-
2|\nabla\nabla\operatorname{Rm}|^2
+
\operatorname{Rm}*\nabla\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
[guided]
The key point of this step is that the Bochner identity is applied to the tensor $A=\nabla\operatorname{Rm}$, not directly to $\operatorname{Rm}$. Let $\pi: M \times I \to M$ denote the projection map $(p,t) \mapsto p$. Define the time-dependent covariant tensor field
\begin{align*}
A \in \Gamma\left((\pi^*T^*M)^{\otimes 5}\right)
\end{align*}
by
\begin{align*}
A=\nabla\operatorname{Rm}.
\end{align*}
For any fixed time $t$, the tensor Bochner identity, equivalently the product rule for the rough Laplacian applied to the pointwise norm of a tensor field, gives
\begin{align*}
\Delta |A|^2
=
2(\Delta A,A)
+
2|\nabla A|^2.
\end{align*}
Here $|\nabla A|^2=|\nabla\nabla\operatorname{Rm}|^2$, because $A=\nabla\operatorname{Rm}$.
There is one extra issue compared with a fixed metric: the norm $|A|^2$ depends on $g(t)$. Differentiating $|A|^2$ in time differentiates both $A$ and the copies of $g(t)^{-1}$ used to contract the tensor indices. Since Ricci flow gives
\begin{align*}
\partial_t g = -2\operatorname{Ric},
\end{align*}
the inverse metric satisfies
\begin{align*}
\partial_t g^{-1}=2\operatorname{Ric}^{\sharp}.
\end{align*}
Here $\operatorname{Ric}^{\sharp}$ is the $(2,0)$-tensor obtained from $\operatorname{Ric}$ by raising both indices with $g(t)^{-1}$. Thus every metric-variation term contains one factor of $\operatorname{Ric}$ and two factors of $A$. Since $\operatorname{Ric}$ is obtained from $\operatorname{Rm}$ by contraction, those terms have the controlled schematic form
\begin{align*}
\operatorname{Ric}*A*A
=
\operatorname{Rm}*A*A.
\end{align*}
Combining the time derivative and the Laplacian identity gives
\begin{align*}
(\partial_t-\Delta)|A|^2
=
2\bigl((\partial_t-\Delta)A,A\bigr)
-
2|\nabla A|^2
+
\operatorname{Rm}*A*A.
\end{align*}
Finally substituting $A=\nabla\operatorname{Rm}$ gives
\begin{align*}
(\partial_t-\Delta)|\nabla\operatorname{Rm}|^2
=
2\bigl((\partial_t-\Delta)\nabla\operatorname{Rm},\nabla\operatorname{Rm}\bigr)
-
2|\nabla\nabla\operatorname{Rm}|^2
+
\operatorname{Rm}*\nabla\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
This is where the desired negative term appears: it is exactly the Bochner term $-2|\nabla A|^2$.
[/guided]
[/step]
[step:Estimate the remaining curvature contractions]
From the commuted evolution equation,
\begin{align*}
(\partial_t-\Delta)\nabla\operatorname{Rm}
=
\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
Hence the inner-product term satisfies
\begin{align*}
2\bigl((\partial_t-\Delta)\nabla\operatorname{Rm},\nabla\operatorname{Rm}\bigr)
=
\operatorname{Rm}*\nabla\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
Combining this with the metric-variation term from the previous step gives
\begin{align*}
(\partial_t-\Delta)|\nabla\operatorname{Rm}|^2
=
-2|\nabla\nabla\operatorname{Rm}|^2
+
\operatorname{Rm}*\nabla\operatorname{Rm}*\nabla\operatorname{Rm}.
\end{align*}
By the definition of controlled contraction notation, there is a constant $C_n>0$, depending only on the dimension $n$, such that
\begin{align*}
\left|\operatorname{Rm}*\nabla\operatorname{Rm}*\nabla\operatorname{Rm}\right|
\leq
C_n|\operatorname{Rm}|\,|\nabla\operatorname{Rm}|^2.
\end{align*}
Therefore,
\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*}
This is the claimed pointwise evolution inequality.
[/step]
