Paper · Apr 20, 2020

Theory and numerics for Chen's flow of curves

Matthew Cooper, Glen Wheeler, Valentina-Mira Wheeler

math.DGWider flowsarXiv:2004.09052
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Abstract

In this article we study Chen's flow of curves from theoreical and numerical perspectives. We investigate two settings: that of closed immersed -circles, and immersed lines satisfying a cocompactness condition. In each of the settings our goal is to find geometric conditions that allow us to understand the global behaviour of the flow: for the cocompact case, the condition is straightforward and the argument is largely standard. For the closed case however, the argument is quite complex. The flow shrinks every initial curve to a point if it does not become singular beforehand, and we must identify a condition to ensure this behaviour as well as identify the point in order to perform the requisite rescaling. We are able to successfully conduct a full analysis of the rescaling under a curvature condition. The analysis resembles the case of the mean curvature flow more than other fourth-order curvature flow such as the elastic flow or the curve diffusion flow, despite the lack of maximum and comparison principles. Our work is informed by a numerical study of the flow, and we include a section that explains the algorithms used and gives some further simulations.

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Submitted
Apr 20, 2020
Subjects
math.DG

Comments: 35 pages, 9 figures

Mean curvature flowOther curvature flowsLimit spaces & RCDNumerics & visualization

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 Mean curvature flow with free boundary - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)6Wheeler 2016 Concentration-compactness and finite-time singularities for Chen's flow (Yann Bernard, Glen Wheeler and Valentina-Mira Wheeler, 2017)1Bernard 2017 On a curvature flow model for embryonic epidermal wound healing (Shuhui He, Glen Wheeler and Valentina-Mira Wheeler, 2018)4He 2018 Higher order curvature flows of plane curves with generalised Neumann boundary conditions (James McCoy, Glen Wheeler and Yuhan Wu, 2020)8McCoy 2020 On the H¹(ds)-gradient flow for the length functional (Philip Schrader, Glen Wheeler and Valentina-Mira Wheeler, 2021)3Schrader 2021 On an inverse curvature flow in two-dimensional space forms (Kwok-Kun Kwong, Yong Wei, Glen Wheeler et al., 2021)7Kwong 2021 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)5Cuthbertson 2024 A curvature flow approach to dorsal closure modelling (Shuhui He, Ben Whale, Glen Wheeler et al., 2025)2He 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Concentration-compactness and finite-time singularities for Chen's flow

    Yann Bernard, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Limit spaces & RCD, Other curvature flows · Same authors Glen Wheeler, Valentina-Mira Wheeler · Shared terms “Chen flow”, “fourth order”, “order curvature”

  2. A curvature flow approach to dorsal closure modelling

    Shuhui He, Ben Whale, Glen Wheeler et al.Wider flows

    Topics Numerics & visualization, Other curvature flows · Same authors Glen Wheeler, Valentina-Mira Wheeler · Shared terms “numerical”, “simulations”

  3. On the -gradient flow for the length functional

    Philip Schrader, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Numerics & visualization · Same authors Glen Wheeler, Valentina-Mira Wheeler · Shared terms “numerical”, “simulations”, “rescaling”

  4. On a curvature flow model for embryonic epidermal wound healing

    Shuhui He, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Other curvature flows · Same authors Glen Wheeler, Valentina-Mira Wheeler · Shared terms “rescaling”

  5. Curve shortening flow with an ambient force field

    Sam Cuthbertson, Glen Wheeler and Valentina-Mira WheelerWider flows

    Same authors Glen Wheeler, Valentina-Mira Wheeler · Shared terms “rescaling”

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

    Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Other curvature flows · Same authors Glen Wheeler, Valentina-Mira Wheeler

  7. On an inverse curvature flow in two-dimensional space forms

    Kwok-Kun Kwong, Yong Wei, Glen Wheeler et al.Wider flows

    Topics Other curvature flows · Same authors Glen Wheeler, Valentina-Mira Wheeler

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

    James McCoy, Glen Wheeler and Yuhan WuWider flows

    Topics Other curvature flows · Same author Glen Wheeler · Shared terms “order curvature”, “initial curve”, “diffusion”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.