one of the following holds:

-   $T < +\infty$ and $C(t)$ shrinks to a point on $\Sigma$ as $t \to T$;

-   $T=+\infty$ and $\ell(t) \to \ell_\infty >0$ as $t \to +\infty$.

As $\ell(t)$ is a non-increasing function of $t$ by Lemma \[L:length-evolution\], it either converges to $0$ or to some positive number $\ell_\infty>0$ as $t \to T$. By short time existence (Theorem \[T:short-time-existence\]), it cannot converge to $\ell_\infty>0$ in finite time. So when $T<+\infty$, $C(t)$ must converge to a point on $\Sigma$ by compactness of $\Sigma$. It remains to show that $\ell(t)$ cannot converge to $0$ if $T=+\infty$. We will prove this by a contradiction argument. Suppose, on the contrary, that $T=+\infty$ and $\ell(t) \to 0$ as $t \to +\infty$. Since $\Sigma$ is compact, there exists some constant $\epsilon_0>0$ such that for any two points $p,q \in \Sigma$ with $d(p,q)<\epsilon_0$, the chord $C_{p,q}$ joining them has $\|\eta^T\|^2_{L^2}$ be bounded from below by a universal positive constant (see, for example, [@Colding-Minicozzi Lemma 5.2]). By Lemma \[L:length-evolution\], $\ell(t)$ must decrease to zero in finite time, which is a contradiction.

Next, we claim that if the flow exists for all time (i.e. $T=+\infty$), then it must converge to an orthogonal geodesic chord 