Paper · Jan 17, 2020

Higher order curvature flows of plane curves with generalised Neumann boundary conditions

James McCoy, Glen Wheeler, Yuhan Wu

math.APWider flowsarXiv:2001.06140
View PDF View HTML arXiv page

As on arXiv

Abstract

We consider the parabolic polyharmonic diffusion and -gradient flows of the -th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in , then the evolving curve converges exponentially in the topology to a straight horizontal line segment. The same behaviour is shown for the -gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on .

View PDF View HTML arXiv page
Submitted
Jan 17, 2020
Subjects
math.AP

Comments: 18 pages, one figure

Other curvature flowsBoundaries & non-compactFree boundary & capillaryLong-time existence & convergence

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 Neumann boundary value problem for gernera curvature flow with forcing term (Ling Xiao, 2016)2Xiao 2016 Mean Curvature Flows of Graphs with Neumann Boundary condition (Jinju Xu, 2016)1Xu 2016 Mean curvature flow with free boundary - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)8Wheeler 2016 A remark on gradient estimates for spacelike mean curvature flow with boundary conditions (Ben Lambert, 2016)6Lambert 2016 Level set mean curvature flow with Neumann boundary conditions (Satoru Aimi, 2021)4Aimi 2021 Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition (Jorge Lira and Fernanda Roing, 2022)3Lira 2022 Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions (Can Cui and Nung Kwan Yip, 2025)7Cui 2025 Area-preserving curvature flow for open curves with general contact angles on skew lines (Fuya Hiroi, 2026)5Hiroi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean Curvature Flows of Graphs with Neumann Boundary condition

    Jinju XuWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  2. Neumann boundary value problem for gernera curvature flow with forcing term

    Ling XiaoWider flows

    Topics Free boundary & capillary, Long-time existence & convergence, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  3. Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition

    Jorge Lira and Fernanda RoingWider flows

    Topics Free boundary & capillary, Long-time existence & convergence, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  4. Level set mean curvature flow with Neumann boundary conditions

    Satoru AimiWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

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

    Fuya HiroiWider flows

    Topics Free boundary & capillary, Long-time existence & convergence, Boundaries & non-compact · Shared terms “higher order”, “initial curve”, “boundary condition”

  6. A remark on gradient estimates for spacelike mean curvature flow with boundary conditions

    Ben LambertWider flows

    Topics Free boundary & capillary, Long-time existence & convergence, Boundaries & non-compact · Shared terms “boundary condition”, “Neumann boundary”

  7. Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

    Can Cui and Nung Kwan YipWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary condition”

  8. Mean curvature flow with free boundary - Type 2 singularities

    Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary, Other curvature flows · Same author Glen Wheeler

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)