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I am reading a book on Ricci flow and at one point there is a computation $\Box^* v$ where

$v=[\tau(2\Delta f - |\nabla f|^2 + R) + f -n]u $

and

$\Box^* = -\partial_t - \Delta + R.$

To compute $\Box^* v$ we split into a product:

$\Box^* v =\Box^* \Bigg(\frac{v}{u}u\Bigg)= \frac{v}{u} \Box^* u \:+\: -u \Bigg(\frac{\partial}{\partial t} + \Delta \Bigg) \Bigg(\frac{v}{u} \Bigg) -2<\nabla \Bigg( \frac{v}{u} \Bigg), \nabla u >$

If you carry out the computation using the definition this seems to imply that $uR\frac{u}{v}$ is equal to the last term, could someone clarify for me why this is the case?

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  • $\begingroup$ Could you say what the identity you’re trying to prove is? And what assumptions if any do $u$ and $v$ satisfy? $\endgroup$
    – Deane Yang
    Commented Mar 2, 2019 at 23:25
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    $\begingroup$ Consider the difference of the left-hand side and the first summand. This consists of terms where at least one derivative hits $v/u$. The second summand is when all derivatives hit $v/u$, and the final one is the cross term when one derivative from $\Delta$ hits $u$ and the other hits $v/u$. $\endgroup$
    – Jeffrey Case
    Commented Mar 3, 2019 at 1:15

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