a point in finite time (by Theorem \[T:contraction\]).

![Any initial chord $C_0$ through the origin (other than the major axis) would converge under the chord shortening flow to the minor axis.[]{data-label="F:ellipse"}](ellipse.png){height="5cm"}

We end this section with a fundamental result on the short time existence and uniqueness for the chord shortening flow.

\[T:short-time-existence\] For any initial chord $C_0:I \to \mathbb{R}^n$ with $C_0(\partial I) \subset \Sigma$, there exists an $\epsilon>0$ and a smooth solution $C(u,t):I \times [0,\epsilon) \to \mathbb{R}^n$ to the chord shortening flow relative to $\Sigma$ as in Definition \[D:CSF\] with initial condition $C(\cdot,0)=C_0$. Moreover, the solution is unique.

Note that for any given $p \neq q \in \Sigma$, the outward unit conormal $\eta$ at the end points $p,q$ of the chord $C_{p,q}$ is given by $$\eta(p)=\frac{p-q}{|p-q|} =-\eta(q).$$ Therefore, Definition \[D:CSF\] (c) is equivalent to the following system of nonlinear system of first order ODEs: $$\label{E:CSF'}
\left\{ \begin{array}{l}
\frac{dp}{dt}=- \frac{\pi_p(p-q)}{|p-q|} \\
\frac{dq}{dt}=- \frac{\pi_q(q-p)}{|q-p|} 
\end{array} \right.$$ where $\pi_x:{\mathbb{R}}^n \to {\mathbb{R}}^n$ is the orthogonal projection onto $T_x\Sigma$ (which depends smoothly on $x$). Since the right hand side of (\[E:CSF’\]) is a Lipschitz function in $p$ and $q$ as long as $|p-q|$ is bounded away from $0$. Therefore, the existence and uniqueness to the 