until they meet at the mid-point of $C(0)$ at the “blow-up” time $T=d(p,q)/2$.

\[ex:annulus\] Let $\Sigma$ be a union of two disjoint circles in ${\mathbb{R}}^2$. We will see (from Theorem \[T:convergence\]) that any chord joining two distinct connected components of $\Sigma$ would evolve under the chord shortening flow to a limit chord $C_\infty$ orthogonal to $\Sigma$ as $t \to \infty$. The same phenomenon holds for any $\Sigma \subset {\mathbb{R}}^n$ which is disconnected.

![A limit chord $C_\infty$ which meets $\partial \Omega$ orthogonally but not lying inside $\Omega$.[]{data-label="F:annulus"}](annulus.png){height="5cm"}

Let $\Sigma$ be the ellipse $\{(x,y) \in {\mathbb{R}}^2 \, : \, x^2+4y^2=1\}$ in ${\mathbb{R}}^2$. By symmetry it is not difficult to see that for any initial chord passing through the origin (with the exception of the major axis), it would evolve under the chord shortening flow to the minor axis of the ellipse, which is a chord orthogonal to $\Sigma$ and contained inside the region enclosed by the ellipse. See Figure \[F:ellipse\]. This example shows that the number of distinct orthogonal chords guaranteed by the Lyusternik-Schnirelmann Theorem is optimal. If we start with an initial chord that lies completely on one side of the major or minor axis, then the chord will shrink to 