As on arXiv
Abstract
We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics. In order to prove the first theorem, we employ the generalization of Grayson's curve shortening flow developed by Angenent-Oaks.
- Submitted
- Feb 2, 2020
- Last revised
- Jan 19, 2022 (v2)
- Subjects
- math.DG, math.DS, math.SG
- Journal
- Journal of Fixed Point Theory and Applications 24 (2022), no. 2, Paper No. 19 (Claude Viterbo's 60th birthday Festschrift)
Comments: 57 pages, 4 figures; final version: a few corrections to Theorem 1.2, to Sections 2.5-6, and to Section 5.4
Its neighbourhood
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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.
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Topics Curve shortening · Same author Marco Mazzucchelli · Shared terms “closed geodesics”, “simple closed”, “infinitely”
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Topics Curve shortening · Same author Marco Mazzucchelli · Shared terms “closed geodesics”, “simple closed”
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Topics Curve shortening · Shared terms “Angenent”
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On the rate of convergence of the -curve shortening flow
Topics Curve shortening
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Self-similar curve shortening flow in hyperbolic 2-space
Topics Curve shortening
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Stabilization technique applied to curve shortening flow in
Topics Curve shortening · Shared terms “developed”
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Sharp distance comparison for curve shortening flow on the round sphere
Topics Curve shortening
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6A.10Curve Shortening and the First Geometric Flows
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.
- 12C.4Loops, Disks and Curve Shrinking
Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.