Paper · Oct 8, 2024
Asymptotic circularity of immortal area-preserving curvature flows
As on arXiv
Abstract
For a class of area-preserving curvature flows of closed planar curves, we prove that every immortal solution becomes asymptotically circular without any additional assumptions on initial data. As a particular corollary, every solution of zero enclosed area blows up in finite time. This settles an open problem posed by Escher–Ito in 2005 for Gage's area-preserving curve shortening flow, and moreover extends it to the surface diffusion flow of arbitrary order. We also establish a general existence theorem for nontrivial immortal solutions under almost circularity and rotational symmetry.
- Submitted
- Oct 8, 2024
- Last revised
- Nov 18, 2024 (v2)
- Subjects
- math.DG, math.AP
Comments: 16 pages, 2 figures, v2: proof of exponential decay corrected
Its neighbourhood
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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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On a non-local area-preserving curve flow
Topics Volume-preserving & constrained flows, Other curvature flows · Shared terms “area preserving”, “preserving curve”, “planar curves”
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Star-shaped Curves under Gage's Area-preserving Flow and the CSF
Topics Curve shortening, Volume-preserving & constrained flows, Singularities · Shared terms “area preserving”, “Gage area”
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Topics Curve shortening, Volume-preserving & constrained flows · Shared terms “area preserving”, “circular”
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Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “area preserving”
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Topics Volume-preserving & constrained flows, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “area preserving”
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Topics Volume-preserving & constrained flows, Long-time existence & convergence, Other curvature flows · Shared terms “area preserving”, “circular”, “initial data”
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Immortal Curve Shortening Flow on Analytic Surfaces
Topics Curve shortening, Long-time existence & convergence · Shared terms “immortal”
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Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus
Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “surface diffusion”, “diffusion flow”
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.