Chord shortening flow and a theorem of Lusternik and Schnirelmann

Martin Man-chun Li Chinese University of Hong Kong

Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:1910.43019

Pacific Journal of Mathematics, 299, (2), 469-488, 2019.5
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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@inproceedings{martin2019chord,
  title={Chord shortening flow and a theorem of Lusternik and Schnirelmann},
  author={Martin Man-chun Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020105913396191548},
  booktitle={Pacific Journal of Mathematics},
  volume={299},
  number={2},
  pages={469-488},
  year={2019},
}
Martin Man-chun Li. Chord shortening flow and a theorem of Lusternik and Schnirelmann. 2019. Vol. 299. In Pacific Journal of Mathematics. pp.469-488. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020105913396191548.
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