Chord Shortening Flow and a Theorem of Lusternik and Schnirelmann

TL;DR

This paper introduces the chord shortening flow, a new geometric flow, and uses it to provide a simplified proof of a classical theorem on the existence of multiple orthogonal geodesic chords, with applications to convex planar domains.

Contribution

The paper presents the chord shortening flow and applies it to give a simplified proof of a classical theorem on geodesic chords, extending previous results.

Findings

Convex chords not orthogonal to the boundary shrink to a point in finite time.

The chord shortening flow is the negative gradient flow of the length functional.

A simplified proof of Lusternik and Schnirelmann's theorem is provided.

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.