initial value problem follows from the fundamental local existence and uniqueness theorem for first order ODE systems (see for example [@Taylor Theorem 2.1]). Hence, (\[E:CSF’\]) is uniquely solvable on some interval $t \in [0,\epsilon)$ for any initial data $p(0)=p_0$ and $q(0)=q_0$ such that $p_0 \neq q_0 \in \Sigma$. Finally we get a solution $C(u,t):I \times [0,\epsilon) \to {\mathbb{R}}^n$ to the chord shortening flow by defining $C(\cdot,t):I \to {\mathbb{R}}^n$ to be the constant speed parametrization of the chord $C_{p_t,q_t}$.

Evolution equations {#S:evolution}
===================

In this section, we derive the evolution of some geometric quantities under the chord shortening flow relative to any $k$-dimensional submanifold $\Sigma$ in ${\mathbb{R}}^n$.

Let $C:I=[0,1] \to {\mathbb{R}}^n$ be a chord joining $p$ to $q$. For any (vector-valued) function $f:\partial I =\{ 0,1\} \to {\mathbb{R}}^m$, we define the *$L^2$-norm* $\|f\|_{L^2}$ and the *sum* $\overline{f}$ of $f$ to be$$\label{E:norm}
\|f\|_{L^2}:=( |f(0)|^2+|f(1)|^2)^{\frac{1}{2}} \qquad \text{and} \qquad \overline{f}:=f(0)+f(1).$$ Also, we define the $\frac{1}{2}$-Laplacian of $f$ relative to the chord $C$ to be the vector-valued function ${\Delta^{\frac{1}{2}}}f:\partial I=\{0,1\} \to {\mathbb{R}}^m$ defined by $$\label{E:DN}
(\Delta^{\frac{1}{2}} f)(0)= \frac{f(0)-f(1)}{\ell} =- (\Delta^{\frac{1}{2}} f)(1),$$ where $\ell=|p-q|$ is the length of the chord $C$.

\[L:DN\] Given any $f:\partial I \to {\mathbb{R}}^m$, we have $\overline{{\Delta^{\frac{1}{2}}}f}=0$ and $\overline{{\langle}f, {\Delta^{\frac{1}{2}}}f {\rangle}}=\frac{\ell}{2} \|{\Delta^{\frac{1}{2}}}f\|^2_{L^2} 