Firstly, I would like to say that I'm studying by myself Riemannian Geometry in order to be able to understand Mean Curvature Flow, so there are some computations that I don't understand well (as I will expain here).
I'm reading Asymptotic behavior for singularities of the mean curvature flow at the moment and I have some doubts about the proof of Huisken's monotonicity formula (Theorem $3.1$ on the article):
1) On the next line of the proof, why the term $\int_{M_t} \rho \frac{1}{2\tau} \langle F, H\nu \rangle d\mu$ change the sign for $+$? I think it's not a typing error, because the sign of this term it's relevant in the final of the proof.
2) How exactly can I compute div $Y$? I know there is a way to compute the divergence on Riemannian manifold given on page $7$ of this notes, but I'm confuse about how compute the divergence of $Y$ using this.
Thanks in advance!
$\textbf{EDIT:}$
I understood why sign changes with respect to my first question. I will put the explanation here
\begin{align} - \int_{M_t} \rho \left\{ H^2 - \frac{n}{2 \tau} + \frac{1}{2\tau} \langle \textbf{F},\textbf{H} \rangle + \frac{|\textbf{F}|^2}{4 \tau^2} \right\} d\mu &= - \int_{M_t} \rho \left\{ - \frac{n}{2 \tau} + H^2 + \frac{1}{2\tau} \langle \textbf{F},\textbf{H} \rangle + \frac{1}{2\tau} \langle \textbf{F},\textbf{H} \rangle - \frac{1}{2\tau} \langle \textbf{F},\textbf{H} \rangle + \frac{|\textbf{F}|^2}{4 \tau^2} \right\} d\mu \\ &= - \int_{M_t} \rho \left\{ - \frac{n}{2 \tau} + H^2 + \frac{2}{2\tau} \langle \textbf{F},\textbf{H} \rangle + \frac{|\textbf{F}|^2}{4 \tau^2} - \frac{1}{2\tau} \langle \textbf{F},\textbf{H} \rangle \right\} d\mu \\ &= - \int_{M_t} \rho \left\{ - \frac{n}{2 \tau} + \left| \textbf{H} + \frac{\textbf{F}}{2\tau} \right|^2 - \frac{1}{2\tau} \langle \textbf{F},\textbf{H} \rangle \right\} d\mu \\ &= - \int_{M_t} \rho \left| \textbf{H} + \frac{\textbf{F}}{2\tau} \right|^2 d\mu + \int_{M_t} \rho \frac{1}{2\tau} \langle \textbf{F},\textbf{H} \rangle d\mu + \int_{M_t} \rho \frac{n}{2 \tau} d\mu. \end{align}