Paper · Dec 24, 2024

Star-shaped Curves under Gage's Area-preserving Flow and the CSF

Laiyuan Gao, Shicheng Zhang, Yuntao Zhang

math.DGWider flowsarXiv:2412.18102
View PDF View HTML arXiv page

As on arXiv

Abstract

Mayer asks a question what closed, embedded and nonconvex initial curves guarantee that Gage's area-preserving flow (GAPF) exists globally. A folklore conjecture since 2012 says that GAPF evolves smooth, embedded and star-shaped initial curves globally. In this paper, we prove this conjecture by using Dittberner's singularity analysis theory. A star-shaped "flying wing" curve is constructed to show that GAPF may not always preserve the star-shapedness of evolving curves. This example is also a negative answer to Mantegazza's open problem whether the curve shortening flow (CSF) always preserves the star shape of the evolving curves.

View PDF View HTML arXiv page
Submitted
Dec 24, 2024
Subjects
math.DG

Comments: 17pages, 4 gigures

Curve shorteningVolume-preserving & constrained flowsSingularities

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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Long-term behavior of curve shortening flow in ℝ³ (Jiří Minarčík and Michal Beneš, 2022)7Minarčík 2022 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)5Litzinger 2023 Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow (Gabriel Khan, 2023)8Khan 2023 Asymptotic circularity of immortal area-preserving curvature flows (Tatsuya Miura, 2024)1Miura 2024 On a non-local area-preserving curve flow (Zezhen Sun and Yuting Wu, 2025)4Sun 2025 On an area-preserving inverse curvature flow for plane curves (Zezhen Sun and Yuting Wu, 2025)6Sun 2025 Shrinkers of the area-preserving curve-shortening flow: Existence and saddle-point property (Nikita Cernomazov, 2025)2Cernomazov 2025 Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds (Ze-Yu Ye and Xiao-Liu Wang, 2026)3Ye 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotic circularity of immortal area-preserving curvature flows

    Tatsuya MiuraWider flows

    Topics Curve shortening, Volume-preserving & constrained flows, Singularities · Shared terms “area preserving”, “Gage area”

  2. Shrinkers of the area-preserving curve-shortening flow: Existence and saddle-point property

    Nikita Cernomazov

    Topics Curve shortening, Volume-preserving & constrained flows · Shared terms “area preserving”

  3. Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds

    Ze-Yu Ye and Xiao-Liu WangWider flows

    Topics Volume-preserving & constrained flows, Singularities · Shared terms “Gage area”, “area preserving”, “initial curves”

  4. On a non-local area-preserving curve flow

    Zezhen Sun and Yuting WuWider flows

    Topics Volume-preserving & constrained flows · Shared terms “area preserving”, “evolving curves”

  5. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening, Singularities · Shared terms “initial curves”

  6. On an area-preserving inverse curvature flow for plane curves

    Zezhen Sun and Yuting WuWider flows

    Topics Volume-preserving & constrained flows · Shared terms “area preserving”, “evolving curves”, “exists globally”

  7. Long-term behavior of curve shortening flow in

    Jiří Minarčík and Michal BenešWider flows

    Topics Curve shortening, Singularities · Shared terms “Gage”

  8. Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

    Gabriel KhanWider flows

    Topics Curve shortening, Singularities

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.

Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    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.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    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.