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I am currently reading Topping's lecture notes on Ricci flow. At one point in the narrative (page 65) he says that using the fact that

$\bigg(\frac{\partial}{\partial t}\nabla - \nabla \frac{\partial}{\partial t} \bigg)A=A\:*\:Rm $

for any time-dependent tensor field $A$, we can check that with respect to local coordinates {$x^{i}$}, the following is true:

$\bigg|\frac{\partial^{l}}{\partial t^{l}}\nabla^{k} \partial_{i} \bigg| \leq C$

for each index $i$ where $\partial_{i}=\frac{\partial}{\partial x^{i}}$ and $C$ is independent of $t$. I am not sure why this follows, could someone clarify?

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  • $\begingroup$ I took a look at Topping's notes, and I don't quite understand the argument, either. But he's trying to prove a standard result, namely that if the curvature tensor remains bounded on a time interval $[0,T)$, then it can be extended to $[0,T+\epsilon)$ for some $\epsilon > 0$. This is a necessary basic first step in the study of the Ricci flow, so I'm sure there are other expositions of it, maybe in one of Ben Chow's books. $\endgroup$
    – Deane Yang
    Commented Feb 18, 2019 at 13:58
  • $\begingroup$ Another approach that should work is the following: First, prove that if the curvature remains bounded on $[0,T)$, so do all of its covariant derivatives. Show that this implies the the flow extends smoothly to $[0,T$. This probably can be done by constructing time-dependent harmonic coordinates that depend smoothly on time. Now solve the Ricci flow on $[T,T+\epsilon)$. Show the extended solution is smooth across $t = T$. $\endgroup$
    – Deane Yang
    Commented Feb 18, 2019 at 14:05
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    $\begingroup$ Finally, I would add that you're probably mostly interested in the Ricci flow for large time, so it's probably best to assume this result and move on. $\endgroup$
    – Deane Yang
    Commented Feb 18, 2019 at 14:06

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