As on arXiv
Abstract
In this paper we consider the anisotropic curve shortening flow in the plane in the presence of an ambient force. We consider force fields in which all their derivatives are bounded in the sense. We prove that closed embedded curves that have a minimum of curvature sufficiently large shrink to round points. The method of proof follows along the same lines of Gage and Hamilton, in that we study a rescaling to prove curvature bounds. We additionally show that the influence of an ambient force field may make such a result untrue, by giving sufficient conditions on the ambient field that ensures eventual non-convexity of an initially convex curve evolving under the flow.
Comments: 37 pages
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 curvature flow that deforms curves to an embedded target
Topics Curve shortening · Same author Glen Wheeler
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Monotonicity of the modulus under curve shortening flow
Topics Curve shortening · Shared terms “curvature bounds”
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Non-uniqueness of curve shortening flow
Topics Curve shortening
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Homogeneous Sobolev gradient flow of the length functional
Topics Curve shortening · Same author Glen Wheeler
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The inverse curve shortening flow on the hyperbolic plane
Topics Curve shortening
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Lectures on curve shortening flow
Topics Curve shortening · Shared terms “embedded curve”, “closed embedded”, “round point”
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Curve Shortening Flow of Space Curves with Convex Projections
Topics Curve shortening
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Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
Topics Curve shortening · Shared terms “eventual”
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.