Paper · Oct 10, 2024
Curve Shortening Flow of Space Curves with Convex Projections
As on arXiv
Abstract
We show that under Space Curve Shortening flow any closed immersed curve in whose projection onto is convex remains smooth until it shrinks to a point. Throughout its evolution, the projection of the curve onto remains convex. As an application, we show that any closed immersed curve in can be perturbed to an immersed curve in whose evolution by Space Curve Shortening shrinks to a point.
Comments: 33 pages, 5 figures. Added affiliation, no other changes
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Singularities of Curve Shortening Flow with Convex Projections
Topics Curve shortening · Same author Qi Sun · Shared terms “convex projection”, “closed immersed”, “immersed curve”
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On asymptotically conical curve shortening flows with convex projections
Topics Curve shortening · Shared terms “convex projection”
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Long-term behavior of curve shortening flow in
Topics Curve shortening · Shared terms “space curves”, “projection”, “remains convex”
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Non-uniqueness of curve shortening flow
Topics Curve shortening
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Topics Curve shortening · Shared terms “space curves”
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Curve shortening flow with an ambient force field
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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.