As on arXiv
Abstract
Space curve motion describes dynamics of material defects or interfaces, can be found in image processing or vortex dynamics. This article analyses some properties of space curves evolved by the curve shortening flow. In contrast to the classical case of shrinking planar curves, space curves do not obey the Avoidance principle in general. They can lose their convexity or develop non-circular singularities even if they are simple. In the first part of the text, we show that even though the convexity of space curves is not preserved during the motion, their orthogonal projections remain convex. In the second part, the Avoidance principle for spherical curves under the curve shortening flow in is shown by generalizing the arguments developed by Hamilton and Gage.
- Submitted
- Dec 22, 2022
- Subjects
- math.DG, math.AP
- Journal
- SIAM Journal on Mathematical Analysis, Vol. 52, Iss. 2 (2020)
Comments: Published in SIAM Journal on Mathematical Analysis, Vol. 52, Iss. 2 (2020)
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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Trajectory Surfaces of Framed Curvature Flow
Topics Curve shortening · Same authors Jiří Minarčík, Michal Beneš
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On asymptotically conical curve shortening flows with convex projections
Topics Curve shortening · Shared terms “long term”, “term behavior”, “projection”
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Curve Shortening Flow of Space Curves with Convex Projections
Topics Curve shortening · Shared terms “space curves”, “projection”, “remains convex”
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Singularities of low entropy high codimension curve shortening flow
Topics Curve shortening, Singularities · Shared terms “space curves”
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Sharp distance comparison for curve shortening flow on the round sphere
Topics Curve shortening, Singularities · Shared terms “spherical curves”
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Singularities of Curve Shortening Flow with Convex Projections
Topics Curve shortening, Singularities · Shared terms “projection”, “circular”
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Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow
Topics Curve shortening, Singularities · Shared terms “contrast”
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Huisken's Distance Comparison Principle in Higher Codimension
Topics Curve shortening, Singularities · Shared terms “projection”, “circular”
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.