Paper · Jan 31, 2017

Singularities of the area preserving curve shortening flow with a Neumann free boundary condition

Elena Mäder-Baumdicker

math.DGWider flowsarXiv:1701.08948v2
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Abstract

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial curves are convex, then an appropriate rescaling at the finite maximal time of existence yields a grim reaper or half a grim reaper as limit flow. We construct examples of initial curves satisfying the mentioned criterion.

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Submitted
Jan 31, 2017
Last revised
Nov 22, 2017 (v2)
Subjects
math.DG, math.AP

Comments: Improved results, simplification of the presentation

Curve shorteningVolume-preserving & constrained flowsSelf-shrinkers & translatorsSingularitiesBoundaries & non-compactFree boundary & capillary

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.

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 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)6Bourni 2022 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)3Langford 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)4Bourni 2024 Star-shaped Curves under Gage's Area-preserving Flow and the CSF (Laiyuan Gao, Shicheng Zhang and Yuntao Zhang, 2024)7Gao 2024 Shrinkers of the area-preserving curve-shortening flow: Classification and saddle-point property (Nikita Cernomazov, 2025)5Cernomazov 2025 High Codimension Curve Shortening Flow with Free Boundary (Huy The Nguyen and Artemis A. Vogiatzi, 2026)2Nguyen 2026 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)1Bourni 2026 Area-preserving curvature flow for open curves with general contact angles on skew lines (Fuya Hiroi, 2026)8Hiroi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows

    Topics Free boundary & capillary, Curve shortening, Singularities · Shared terms “convex domains”, “finite time”

  2. High Codimension Curve Shortening Flow with Free Boundary

    Huy The Nguyen and Artemis A. VogiatziWider flows

    Topics Free boundary & capillary, Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

  3. A distance comparison principle for curve shortening flow with free boundary

    Mat Langford and Jonathan J. ZhuWider flows

    Topics Free boundary & capillary, Curve shortening, Boundaries & non-compact · Shared terms “boundary condition”, “finite time”

  4. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows

    Topics Free boundary & capillary, Curve shortening · Shared terms “convex domains”

  5. Shrinkers of the area-preserving curve-shortening flow: Classification and saddle-point property

    Nikita Cernomazov

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

  6. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “grim reaper”

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

    Laiyuan Gao, Shicheng Zhang and Yuntao ZhangWider flows

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

  8. Area-preserving curvature flow for open curves with general contact angles on skew lines

    Fuya HiroiWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Boundaries & non-compact · Shared terms “area preserving”, “initial curve”, “boundary condition”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, 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.