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Chord Shortening Flow and a Theorem of Lusternik and Schnirelmann. (arXiv:1707.04014v1 [math.DG])
来源于:arXiv
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. 查看全文>>