0.1] for curve shortening flow on surfaces. For simplicity, we assume that $\Sigma$ is compact. However, the same result holds for non-compact $\Sigma$ which satisfies some convexity condition at infinity as in [@Grayson89].

\[T:convergence\] Let $\Sigma \subset \mathbb{R}^n$ be a compact $k$-dimensional smooth submanifold without boundary. Suppose $C(0):I \to \mathbb{R}^n$ is a chord with distinct end points on $\Sigma$. Then there exists a maximally defined smooth family of chords $C(t):I \to \mathbb{R}^n$ for $t \in [0,T)$ with distinct end points on $\Sigma$, and $C(t)=C(\cdot,t)$ where $C(u,t):I \times [0,T) \to {\mathbb{R}}^n$ is the unique solution to the chord shortening flow (relative to $\Sigma$) as in Definition \[D:CSF\].

Moreover, if $T < + \infty$, then $C(t)$ converges to a point on $\Sigma$ as $t \to T$. If $T$ is infinite, then $C(t)$ converges to an orthogonal geodesic chord with end points on $\Sigma$ as $t \to \infty$.

By the short time existence and uniqueness theorem (Theorem \[T:short-time-existence\]), the chord shortening flow continues to exist and is unique as long as $\ell >0$. Therefore, $C(t)$ is uniquely defined for $t \in [0,T)$ where either $T < +\infty$ or $T=+\infty$.

\[L:ell-limit\] Let $C(t)$, $t \in [0,T)$, be a maximally defined chord shortening flow. Then 