\frac{\partial}{\partial t} \left( -\frac{1}{\ell} \right) \sum_{i=1}^k \langle \frac{\partial }{\partial u}, e_i \rangle e_i- \frac{1}{\ell} \sum_{i=1}^k \frac{\partial }{\partial t} \left( \langle \frac{\partial }{\partial u},e_i \rangle  e_i \right)\\
&=& \frac{1}{\ell} \|\eta^T\|^2_{L^2} \eta^T - \frac{1}{\ell} \sum_{i=1}^k \left( \langle D_{\frac{\partial}{\partial u}} \frac{\partial}{\partial t}, e_i \rangle e_i+ \langle \frac{\partial}{\partial u}, D_{\frac{\partial}{\partial t}} e_i \rangle e_i  + \langle \frac{\partial }{\partial u},e_i \rangle D_{\frac{\partial}{\partial t}} e_i \right) \\
&=& \frac{1}{\ell} \|\eta^T\|^2_{L^2} \eta^T - \frac{\eta^T}{\ell} - A(\eta^T,\eta^T) - \frac{1}{\ell} \sum_{i=1}^k \left( {\langle}-\eta^T(1),e_i {\rangle}e_i + \ell \langle \eta^N, A(\eta^T,e_i) \rangle e_i  \right) \\
&=&  - {\Delta^{\frac{1}{2}}}\eta^T + \frac{1}{\ell} \|\eta^T\|^2_{L^2} \eta^T  - \sum_{i=1}^k {\langle}A(\eta^T,e_i),\eta^N {\rangle}e_i - \frac{1}{\ell}  (\overline{\eta^T}-\eta^T)^N - A(\eta^T,\eta^T).\end{aligned}$$ A similar calculation yields (\[E:eta-T-evolution\]) at $u=1$. This proves the proposition.

Under the same assumptions as in Theorem \[T:eta-T-evolution\], we have $$\label{E:eta-norm-evolution}
\frac{1}{2} \frac{d}{dt} \|\eta^T\|^2_{L^2}= - \frac{\ell}{2}  \| {\Delta^{\frac{1}{2}}}\eta^T \|^2_{L^2}  + \frac{1}{\ell} \|\eta^T\|^4_{L^2} -\overline{{\langle}A(\eta^T,\eta^T), \eta {\rangle}}.$$

Using (\[E:eta-T-evolution\]) and Lemma \[L:DN\], noting that the last two terms of (\[E:eta-T-evolution\]) are normal to $\Sigma$, we have $$\frac{1}{2} \frac{d}{dt} \|\eta^T\|^2_{L^2} = \overline{{\langle}\eta^T, \frac{\partial \eta^T}{\partial t} {\rangle}} 
= - \frac{\ell}{2}  \| {\Delta^{\frac{1}{2}}}\eta^T \|^2_{L^2}  + \frac{1}{\ell} \|\eta^T\|^4_{L^2} -\overline{{\langle}A(\eta^T,\eta^T), \eta^N {\rangle}}.$$

\[ex:flat2\] In the case of Example \[ex:flat\], we have $\eta^T(0)=-\eta^T(1)$ equals to a constant unit vector independent of $t$ and hence both sides are 