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This question is from my question on mathematics.

Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I can't see $Q\ge 0$ when $t=0$.

Besides, how to ensure $$ \left\{ \begin{aligned} & (\partial_t -\Delta) W_a =\frac{1}{t}W_a \\ & \nabla_aW_b=0 \end{aligned} \right.~~~~~~~~~~~~~ \left\{ \begin{aligned} & (\partial_t-\Delta)U_{ab} = 0 \\ & \nabla_aU_{bc} = \frac{1}{2}(R_{ab}W_c-R_{ac}W_b)+\frac{1}{4t}(g_{ab}W_c-g_{ac}W_b) \end{aligned} \right. $$ have solutions ? And $W_a$ and $U_{ab}$ are arbitrary , how to explain it ? (It is enough that proving the basis meets the equations ?)

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    $\begingroup$ Why don't you look at proper textbooks on the Ricci flow (e.g. one of those by Ben Chow et al.) or the original paper of Hamilton? This reference only gives a sketch, which seems to be extremely poorly written. $\endgroup$
    – YangMills
    Commented May 26, 2016 at 13:36
  • $\begingroup$ @YangMills Do you mean Bennett Chow's Hamilton's Ricci Flow ? $\endgroup$
    – Farmer
    Commented May 30, 2016 at 1:23
  • $\begingroup$ The book by Chow-Lu-Ni "Hamilton's Ricci Flow" has a detailed sketch of the proof. For a complete proof see either Hamilton's original paper, or the book "The Ricci Flow: Techniques and applications, part II: analytic aspects" by Chow-Chu-Glickenstein-Guenther-Isenberg-Ivey-Knopf-Lu-Luo-Ni. $\endgroup$
    – YangMills
    Commented Jun 1, 2016 at 16:36

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