to $\Sigma$ as $t \to \infty$. Since $|\eta^T| \leq \|\eta^T\|_{L^2}$, it suffices to prove the following lemma. Theorem \[T:convergence\] clearly follows from Lemma \[L:ell-limit\] and \[L:kto0\].

\[L:kto0\] Under the same assumption as Lemma \[L:ell-limit\] and suppose $T=+\infty$, then $\|\eta^T\|_{L^2} \to 0$ as $t \to +\infty$.

Write $\ell_t=\ell(t)$ for $t \in [0,+\infty]$. By Lemma \[L:length-evolution\] and \[L:ell-limit\], we have $$\label{E:ell-monotone}
\ell_0 \geq \ell_t \geq \ell_\infty>0 \quad \text{ for all $t$}.$$ Moreover, integrating the inequality in Lemma \[L:length-evolution\] we obtain $$\ell_t- \ell_\infty = \int_{t}^\infty \|\eta^T\|^2_{L^2} \; d\tau \geq 0.$$ As a result, $\int_t^\infty \|\eta^T\|^2_{L^2} \; d\tau \to 0$ as $t \to \infty$. In other words, $\|\eta^T\|^2_{L^2}$ is $L^2$-integrable on $t \in [0,+\infty)$. If we can control the time derivative of $\|\eta^T\|_{L^2}^2$, then we can conclude that $\|\eta^T\|_{L^2} \to 0$ as $t \to \infty$. Using (\[E:eta-norm-evolution\]), (\[E:ell-monotone\]), Lemma \[L:DN\] and $\|\eta^T \|^2_{L^2} \leq 2$, we have the following differential inequality $$\label{E:norm-ineq} 
\frac{1}{2}\frac{d}{dt} \|\eta^T\|^2_{L^2} \leq \left( C+\frac{4}{\ell_\infty} \right) \|\eta^T\|^2_{L^2}$$ where $C=\sup_{\Sigma} |A| > 0$ is a constant depending only on the compact submanifold $\Sigma$. We now combine (\[E:norm-ineq\]) with the fact that $\int_t^\infty \|\eta^T\|^2_{L^2} \; d\tau \to 0$ as $t \to \infty$ to conclude that $\|\eta^T\|^2_{L^2} \to 0$ as $t \to \infty$.

To simplify 