flow as an evolution equation for the two end points lying on $\Sigma$. As a result, the flow is a *non-local* evolution of a pair of points on $\Sigma$ as it depends on the chord joining them. In fact, the chord shortening flow can be regarded as the heat equation for the half-Laplacian (or the *Dirichlet-to-Neumann map*).

The organization of this paper is as follows. In Section 2, we introduce the chord shortening flow, give a few examples, and prove the short time existence and uniqueness of the flow. In Section 3, we derive the evolution equations for some geometric quantities under the chord shortening flow. In Section 4, we prove the long time existence to the flow provided that it does not shrink the chord to a point in finite time. In Section 5, we prove that an initial convex chord inside a compact convex domain in ${\mathbb{R}}^2$ would shrink to a point in finite time under the flow, provided that the initial chord is not an orthogonal geodesic chord.

**Acknowledgement.** The author would like to express his gratitude to Prof. Richard Schoen for his interest in this work. He also want to thank Mario Micallef and Luen-Fai Tam 