Vol. 299, No. 2, 2019

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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
Authors
Martin Man-chun Li
Department of Mathematics
The Chinese University of Hong Kong
Hong Kong
China