Paper · Mar 23, 2020
The curve shortening flow in the metric-affine plane
As on arXiv
Abstract
We investigate for the first time the curve shortening flow in the metric-affine plane and prove that under simple geometric condition it shrinks a closed convex curve to a "round point" in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in Euclidean plane.
- Submitted
- Mar 23, 2020
- Subjects
- math.DG
Comments: 12 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 2016, plus the canon.
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Affine Self-similar solutions of the affine curve shortening flow I: The degenerate case
Topics Curve shortening · Shared terms “Euclidean plane”
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Topics Curve shortening, Singularities · Shared terms “Euclidean plane”, “Gage”, “finite time”
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Topics Curve shortening · Shared terms “convex curve”, “round point”, “Gage”
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Topics Curve shortening, Singularities · Shared terms “round point”
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Topics Curve shortening
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Topics Curve shortening, Singularities · Shared terms “round point”, “finite time”, “Euclidean”
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Topics Curve shortening, Singularities · Shared terms “finite time”
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Topics Curve shortening, Singularities · Shared terms “round point”, “finite time”
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.