for helpful comments and discussions. These work are partially supported by a research grant from the Research Grants Council of the Hong Kong Special Administrative Region, China \[Project No.: CUHK 14323516\] and CUHK Direct Grant \[Project Code: 3132705\].

*Notations.* Throughout this paper, we will denote $I:=[0,1]$ with $\partial I=\{0,1\}$. The Euclidean space ${\mathbb{R}}^n$ is always equipped with the standard inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot |$. For any subset $S \subset {\mathbb{R}}^n$, we use $d(\cdot, S)$ to denote the distance function from $S$.

Chord Shortening Flow {#S:CSF}
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In this section, we introduce a new geometric flow called *chord shortening flow*. This flow has some similarities with the classical curve shortening flow. The main result in this section is the short-time existence and uniqueness theorem for the chord shortening flow (Theorem \[T:short-time-existence\]). We also study some basic examples of such a flow. Let $\Sigma$ be a $k$-dimensional smooth submanifold [^1] in $\mathbb{R}^n$. Note that $\Sigma$ can be disconnected in general. For any two points $p,q \in \Sigma$, we can consider the extrinsic chord distance between them in $\mathbb{R}^n$.

\[D:chord-distance\] The *chord distance function* $d: \Sigma \times \Sigma \to \mathbb{R}_{\geq 0}$ is defined to be $$d(p,q):=\textrm{dist}_{\mathbb{R}^n} (p,q)=|p-q|.$$

Since 