notation, let $f(t):=\|\eta^T\|^2_{L^2}$ and $c:=C+\frac{4}{\ell_\infty}$. Then $\int_t^\infty f \to 0$ as $t \to \infty$ and $f' \leq cf$. We argue that $f(t) \to 0$ as $t \to \infty$. Suppose not, then there exists an increasing sequence $t_n \to +\infty$ such that $$\label{E:conv}
f(t_n) > \frac{1}{n} \qquad \text{and} \qquad \int_{t_n/2}^\infty f \leq \frac{1}{n^3}.$$ We claim that there exists $t_n^* \in  (t_n -\frac{1}{n},t_n+\frac{1}{n})$ such that $f(t_n^*) \leq 1/n^2$. If not, then by (\[E:conv\]) $$\frac{2}{n^3} \leq \int_{t_n-\frac{1}{n}}^{t_n+\frac{1}{n}} f \leq \int_{t_n/2}^\infty f \leq \frac{1}{n^3},$$ which is a contradiction. Using that $f' \leq c f$, we see that by (\[E:conv\]) $$\frac{1}{n} < f(t_n) \leq f(t_n^*) e^{\frac{c}{n}} \leq \frac{1}{n^2} e^{\frac{c}{n}}.$$ As a result, there is a contradiction when $n$ is sufficiently large. We have thus proved that $f(t) \to 0$ as $t \to \infty$, as claimed.

Existence of Orthogonal Geodesic Chords {#S:applications}
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In this section, we give several geometric applications of the chord shortening flow concerning the existence of multiple orthogonal geodesic chords. We first give the precise definition.

\[D:OGC\] Let $\Sigma \subset {\mathbb{R}}^n$ be a smooth $k$-dimensional submanifold without boundary. An *orthogonal geodesic chord for $\Sigma$* is a geodesic $c:[0,1] \to {\mathbb{R}}^n$ with endpoint $c(0)$ and $c(1)$ lying on $\Sigma$ such that $c'(0)$ and 