Paper · Oct 4, 2023
Sharp distance comparison for curve shortening flow on the round sphere
As on arXiv
Abstract
We prove that curve shortening flow on the round sphere displays sharp chord-arc improvement, precisely as in the planar setting (Andrews and Bryan, Comm. Anal. Geom., 2011). As in the planar case, the sharp estimate implies control on the curvature, resulting in a direct and efficient proof that simple spherical curves either contract to round points (in finite time) or converge to great circles (in infinite time).
Its neighbourhood
Connected papers
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 2021, plus the canon.
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A distance comparison principle for curve shortening flow with free boundary
Topics Curve shortening, Singularities · Same authors Mat Langford, Jonathan J. Zhu · Shared terms “distance comparison”, “chord arc”, “infinite time”
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Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow
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Topics Curve shortening, Singularities · Shared terms “round point”, “finite time”
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Topics Curve shortening, Singularities · Shared terms “spherical curves”
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Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
Topics Curve shortening, Singularities · Shared terms “Geom”
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.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.