Paper · Feb 12, 2018

On a curvature flow model for embryonic epidermal wound healing

Shuhui He, Glen Wheeler, Valentina-Mira Wheeler

math.DGWider flowsarXiv:1802.03902
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As on arXiv

Abstract

The paper studies a curvature flow linked to the physical phenomenon of wound closure. Under the flow we show that a closed, initially convex or close-to-convex curve shrinks to a round point in finite time. We also study the singularity, showing that the singularity profile after continuous rescaling is that of a circle. We additionally give a maximal time estimate, with an application to the classification of blowups.

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Submitted
Feb 12, 2018
Subjects
math.DG

Comments: 43 pages

Other curvature 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 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)3Wheeler 2016 Concentration-compactness and finite-time singularities for Chen's flow (Yann Bernard, Glen Wheeler and Valentina-Mira Wheeler, 2017)2Bernard 2017 Theory and numerics for Chen's flow of curves (Matthew Cooper, Glen Wheeler and Valentina-Mira Wheeler, 2020)4Cooper 2020 On the H¹(ds)-gradient flow for the length functional (Philip Schrader, Glen Wheeler and Valentina-Mira Wheeler, 2021)7Schrader 2021 On an inverse curvature flow in two-dimensional space forms (Kwok-Kun Kwong, Yong Wei, Glen Wheeler et al., 2021)6Kwong 2021 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)5Cuthbertson 2024 A curvature flow that deforms curves to an embedded target (Samuel Cuthbertson, Glen Wheeler and Valentina Wheeler, 2024)8Cuthbertson 2024 A curvature flow approach to dorsal closure modelling (Shuhui He, Ben Whale, Glen Wheeler et al., 2025)1He 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A curvature flow approach to dorsal closure modelling

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

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

  2. Concentration-compactness and finite-time singularities for Chen's flow

    Yann Bernard, Glen Wheeler and Valentina-Mira WheelerWider flows

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

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

    Glen Wheeler and Valentina-Mira WheelerWider flows

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

  4. Theory and numerics for Chen's flow of curves

    Matthew Cooper, 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 “convex curve”, “round point”, “rescaling”

  6. 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 · Shared terms “convex curve”

  7. On the -gradient flow for the length functional

    Philip Schrader, Glen Wheeler and Valentina-Mira WheelerWider flows

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

  8. A curvature flow that deforms curves to an embedded target

    Samuel Cuthbertson, Glen Wheeler and Valentina WheelerWider flows

    Topics 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) 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.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.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.