As on arXiv
Abstract
Given two curves bounding a region of area that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time . We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an function.
Comments: v1 was not submitted. v2 has numerous adjustments and improvements to the exposition. v3 has further adjustments and improvements
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 delayed interior area-to-height estimate for the Curve Shortening Flow
Topics Curve shortening · Same author Arjun Sobnack · Shared terms “delayed parabolic”, “parabolic regularity”, “Graphical Curve”
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The Harnack inequality without convexity for curve shortening flow
Topics Curve shortening · Same authors Arjun Sobnack, Peter M. Topping · Shared terms “delayed parabolic”, “parabolic regularity”, “graphical”
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Monotonicity of the modulus under curve shortening flow
Topics Curve shortening · Same authors Arjun Sobnack, Peter M. Topping
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Non-uniqueness of curve shortening flow
Topics Curve shortening · Shared terms “flow starting”
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The inverse curve shortening flow on the hyperbolic plane
Topics Curve shortening
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On asymptotically conical curve shortening flows with convex projections
Topics Curve shortening
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Stabilization technique applied to curve shortening flow in
Topics Curve shortening
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Compact curve shortening flow solutions out of non compact curve
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.