Paper · Jul 16, 2025

A curvature flow approach to dorsal closure modelling

Shuhui He, Ben Whale, Glen Wheeler, Valentina-Mira Wheeler

math.APWider flowsarXiv:2507.12088
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Abstract

In this paper we propose and study a curvature-based mathematical model for dorsal closure in embryonic drosophila. Using an analysis that mixes maximum-principle and integral-estimates, we establish global existence and convergence for data that mimics the initial geometry of a dorsal closure event. Further, we present a numerical approximation scheme for the flow, establishing stability, consistence, and convergence. We also give sample simulations of the flow with initial configurations that include experimentally observed data.

Comments: 39 pages, 4 figures, 3 tables

Other curvature flowsStability & uniquenessLong-time existence & convergenceNumerics & 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 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 Discrete hyperbolic curvature flow in the plane (Klaus Deckelnick and Robert Nürnberg, 2022)6Deckelnick 2022 Stability of piecewise flat Ricci flow (Rory Conboye, 2023)1Conboye 2023 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)8Garfinkle 2024 Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus (Roman Bahadursingh, 2024)4Bahadursingh 2024 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)2Cuthbertson 2024 Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group (Giovanna Citti, Nicolas Dirr, Federica Dragoni et al., 2024)3Citti 2024 A curvature flow that deforms curves to an embedded target (Samuel Cuthbertson, Glen Wheeler and Valentina Wheeler, 2024)5Cuthbertson 2024 Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick (Klaus Deckelnick and Vanessa Styles, 2026)7Deckelnick 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 piecewise flat Ricci flow

    Rory Conboye PDF

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical”, “simulations”, “approximation”

  2. Curve shortening flow with an ambient force field

    Sam Cuthbertson, Glen Wheeler and Valentina-Mira WheelerWider flows PDF

    Same authors Glen Wheeler, Valentina-Mira Wheeler

  3. Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

    Giovanna Citti, Nicolas Dirr, Federica Dragoni et al.Wider flows PDF

    Topics Numerics & visualization, Other curvature flows · Shared terms “mathematical”, “approximation”

  4. Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

    Roman BahadursinghWider flows PDF

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “simulations”, “numerical”

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

    Samuel Cuthbertson, Glen Wheeler and Valentina WheelerWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Same author Glen Wheeler

  6. Discrete hyperbolic curvature flow in the plane

    Klaus Deckelnick and Robert NürnbergWider flows PDF

    Topics Numerics & visualization, Other curvature flows · Shared terms “simulations”, “numerical”, “approximation”

  7. Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick

    Klaus Deckelnick and Vanessa StylesWider flows PDF

    Topics Numerics & visualization · Shared terms “numerical”, “observed”, “approximation”

  8. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al. PDF

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “simulations”, “numerical”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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) 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.