Vtgyjfy

I'm reading a master thesis about Mean Curvature Flow and I'm trying understand how was developed the following equation:

$$frac{partial Gamma^i_{jk}}{partial t} = nabla A ast A.$$

It's how the equation above was developed:

I didn't understand how the terms highlighted appears. I know that $A ast nabla A$ means that I'm considering the linear combination of the contraction of $A$ and $nabla A$ with respect to the metric $g$ (as you can read on the beginning of the section $13$ on page $40$ of Hamilton's article), but I can't see why the line above $(3)$ it's a linear combination of contractions of $nabla H, A, H$ and $nabla A$ and why $nabla H ast A + H ast nabla A = nabla A ast A$. I think if I have more details about the operator $ast$, then I will be able to understand $(3)$.

Thanks in advance!

$textbf{EDIT:}$

I understood how obtain $(1)$ and (2) and I have an idea for answer my doubt
about $(3)$, but I couldn't understand how obtain $(3)$ yet.

$(1)$: using the fact that $frac{partial g^{il}}{partial t} = 2H h^{il}$, we observe that

$$frac{partial g^{il}}{partial t} = frac{partial (g^{is}g_{sz}g^{zl})}{partial t} = frac{partial (g^{is}g^{zl})}{partial t} g_{sz} + (g^{is}g^{zl}) frac{partial g_{sz}}{partial t}$$

$$ = left( frac{partial g^{is}}{partial t} g^{zl} + g^{is} frac{partial g^{zl}}{partial t} right) g_{sz} + (g^{is}g^{zl}) frac{partial g_{sz}}{partial t}$$

$$ = left( 2H h^{is} g^{zl} + g^{is} 2H h^{zl} right) g_{sz} + (g^{is}g^{zl}) frac{partial g_{sz}}{partial t}$$

$$= 2H h^{is} g^l_s + g^i_z 2H h^{zl} + (g^{is}g^{zl}) frac{partial g_{sz}}{partial t}$$

$$= 2H h^{il} + 2H h^{il} + (g^{is}g^{zl}) frac{partial g_{sz}}{partial t}$$

$$=2 (2H h^{il}) + g^{is} frac{partial (g_{sz})}{partial t} g^{zl}$$

$$=2 (frac{partial g^{il}}{partial t}) + g^{is} frac{partial (g_{sz})}{partial t} g^{zl}$$

$$Longrightarrow frac{partial g^{il}}{partial t} = - g^{is} frac{partial (g_{sz})}{partial t} g^{zl}$$

$(2)$: it's just observe that

$$Gamma^z_{jk} = g^{zl} left( frac{partial g_{kl}}{partial x_j} + frac{partial g_{jl}}{partial x_k} - frac{partial g_{jk}}{partial x_l} right)$$

$(3)$: Computing in normal coordinates, we observe that

$(circ) - H(nabla_j h^i_k + nabla_k h^i_j - nabla^i h_{jk}) = - g^{ij}h_{ij}(nabla_j h^i_k + nabla_k h^i_j - nabla^i h_{jk})$

$(circ circ) -h^i_k nabla_j H -h^i_j nabla_k H + h_{jk} nabla^i H = -h^i_k nabla_j (g^{rs}h_{rs}) -h^i_j nabla_k (g^{rs}h_{rs}) + h_{jk} nabla^i (g^{rs}h_{rs}) = -h^i_k left( g^{rs} nabla_j h_{rs} right) -h^i_j left( g^{rs} nabla_k h_{rs} right) + h_{jk} left( g^{rs} nabla^i h_{rs} right)$,

I don't be sure if I'm using the definition of $ast$ correctly, but if I'm not wrong, then $(circ)$ it's a linear combination which terms involve the trace of the tensor $ A$ and the components of the tensor $nabla A$ (it's the components of the tensor $nabla A$ if we rewrite $nabla_j h^i_k = g^{is} nabla_j h_{sk}$ and $nabla_k h^i_j = g^{is} nabla_k h_{sj}$), while $(circ circ)$ it's a linear combination which terms involve the trace of the tensor $nabla A$ and the components of the tensor $A$ (again, it's the components of the tensor $A$ if we rewrite $h^i_k = g^{is}h_{sk}$ and $h^i_j = g^{is}h_{sj}$) nad this justify why $(circ) + (circ circ) = nabla A ast A$, but this not answer why
$(circ) + (circ circ) = nabla H ast A + H ast nabla A$, which lead us to my question:

let be $T$ and $S$ $(0,2)$-tensors (just for simplicity) with components $T_{ij}$ and $S_{kl}$ and denote by $text{tr}_g T$ the trace of the tensor $T$ with respect to the metric $g$. Is it $T ast S$ a linear combination with terms like $text{tr}_g T S_{ij}$, $T_{ij} text{tr}_g S$, $text{tr}_g T text{tr}_g S$ or it's a linear combination with other kind of terms?