Chord shortening flow and a theorem of Lusternik and Schnirelmann

Martin Man-chun Li

Vol. 299 (2019), No. 2, 469–488

DOI: 10.2140/pjm.2019.299.469

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 not orthogonal to the boundary would shrink to a point in finite time under the flow.

Keywords

geometric flows, orthogonal geodesic chords, Lusternik–Schnirelmann theory

Mathematical Subject Classification 2010

Primary: 53C22

Secondary: 58E10

Milestones

Received: 6 February 2018

Revised: 17 June 2018

Accepted: 17 July 2018

Published: 21 May 2019