---
abstract: 'We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord which is not orthogonal to the boundary would shrink to a point in finite time under the flow.'
address: 'Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong'
author:
- 'Martin Man-chun Li'
bibliography:
- 'references.bib'
title: Chord Shortening Flow and a Theorem of Lusternik and Schnirelmann
---

Introduction {#S:intro}
============

The existence of closed geodesics in a Riemannian manifold is one of the most fundamental questions in geometry that has been studied extensively since the time of Poincaré [@Poincare04]. The critical point theories developed by Morse and Lusternik-Schnirelmann have played an essential role in this problem in the early 20th century (see [@Klingenberg] for a detailed exposition up to 1978). Although there does 