relative to point curves on $\Sigma$. Interested readers can refer to [@Gluck-Ziller83] or [@Giannoni-Majer97] for more details (for Theorem \[T:LS\] there is a more elementary proof - see [@Kuiper64] for example).

In this paper, we introduce a new curve shortening process called the *chord shortening flow* (see Definition \[D:CSF\]), which evolves a geodesic chord according to the “contact angle” that the chord makes with $\Sigma$ at its end points. It is the negative gradient flow for the length functional on the space of chords. We study the fundamental properties including the short-time existence and uniqueness and long-time convergence of the flow when the ambient space is ${\mathbb{R}}^n$. Note that the flow still makes sense in certain Riemannian manifolds but for simplicity we postpone the details to another forthcoming paper. The chord shortening flow, as a negative gradient flow, clearly satisfies all the properties (1) - (3) above; hence provide the most natural curve shortening process required in the proof of Theorem \[T:LS\] and \[T:double-normals\] (but not Theorem \[T:Bos\] in its full generality).

We would like to mention that Lusternik and Schnirelmann used the same ideas to prove the *Theorem of Three Geodesics* which asserts that any Riemannian sphere $(S^2,g)$ contains 