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In the book on Ricci flow by Andrews and Hopper, it has been proved that if Ricci flow on $M$ has a finite time singularity at time $T$ then $\lim_{t \nearrow T} \sup_{x\in M} |Rm(x,t)|=\infty$. I am wondering whether the following assertions are false or unknown (I know that the following assertions are true if $\lim$ is replaced by $\lim \sup$):

  1. $\lim_{t \nearrow T} \sup_{x\in M} |Ric(x,t)|=\infty$
  2. $\lim_{t \nearrow T} \sup_{x\in M} |R(x,t)|=\infty$ when $M$ is K\"{a}hler.

Thanks in advance.

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Assertion (1) holds; that is, $Ric$ remains bounded as long as the Ricci flow exists. This was proved by N. Sesum (AJM, 2005), and improves the earlier result of R. Hamilton mentioned in the question which guarantees that $Rm$ stays bounded while the flow exists.

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  • $\begingroup$ Does not $Ric$ is unbounded on $M \times [0,T)$ imply $\lim \sup_{t \nearrow T} \sup_{x\in M} |Ric(x,t)|=\infty$ ? $\endgroup$
    – Bingo
    Mar 1, 2017 at 6:44
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    $\begingroup$ @Bingo: The statement (similarly to Hamilton's) is that if $Ric$ is bounded, then Ricci flow can continue for a little bit longer. Thus, if a finite time singularity is reached, then $Ric$ must blow up. $\endgroup$ Mar 1, 2017 at 14:48

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