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Chord shortening flow
and a theorem of Lusternik and Schnirelmann
Martin Man-chun Li |
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Vol. 299 (2019), No. 2, 469–488
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DOI: 10.2140/pjm.2019.299.469
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Abstract |
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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. |
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Keywords
geometric flows, orthogonal geodesic chords,
Lusternik–Schnirelmann theory
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Mathematical Subject Classification 2010
Primary: 53C22
Secondary: 58E10
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Milestones
Received: 6 February 2018
Revised: 17 June 2018
Accepted: 17 July 2018
Published: 21 May 2019
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Authors |
| Martin Man-chun Li | |
| Department of Mathematics The Chinese University of Hong Kong Hong Kong China |
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