Paper · May 17, 2024
Uniqueness of tangent flows at infinity for finite-entropy shortening curves
As on arXiv
Abstract
In this paper, we prove that an ancient smooth curve shortening flow with finite-entropy embedded in has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplity exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.
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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 2021, plus the canon.
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Classification of ancient finite-entropy curve shortening flows
Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Same authors Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su · Shared terms “finite entropy”, “ancient smooth”, “static line”
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Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Shared terms “grim reaper”
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Generalized Elastic Translating Solitons
Topics Curve shortening, Self-shrinkers & translators · Shared terms “reaper curves”, “grim reaper”
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Topics Curve shortening, Ancient solutions, Self-shrinkers & translators
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Ancient curve shortening flow in the disc with mixed boundary condition
Topics Curve shortening, Ancient solutions, Self-shrinkers & translators · Shared terms “grim reaper”
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A Compact Non-convex Ancient Curve Shortening Flow
Topics Curve shortening, Ancient solutions
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
- 12B.3The Structure of κ-Solutions
Necks and caps, compactness, and universal estimates.
- 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.