Paper · Apr 27, 2022

Discrete hyperbolic curvature flow in the plane

Klaus Deckelnick, Robert Nürnberg

math.NAWider flowsarXiv:2204.12878v3SIAM J. Numer. Anal. 61 (2023) 1835--1857
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Abstract

Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave phenomena in solid-liquid interfaces. We introduce a semidiscrete finite difference method for the approximation of hyperbolic curvature flow and prove error bounds for natural discrete norms. We also present numerical simulations, including the onset of singularities starting from smooth strictly convex initial data.

Comments: 23 pages, 10 figures

Curve shorteningOther curvature flowsSingularitiesNumerics & visualization

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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 A novel finite element approximation of anisotropic curve shortening flow (Klaus Deckelnick and Robert Nürnberg, 2021)1Deckelnick 2021 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)8Litzinger 2023 The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature (Patrick Guidotti, 2023)4Guidotti 2023 Which shapes can appear in a Curve Shortening Flow Singularity? (Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al., 2024)3Angenent 2024 A curvature flow approach to dorsal closure modelling (Shuhui He, Ben Whale, Glen Wheeler et al., 2025)6He 2025 Geometric aspects of the curve shortening flow in the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)7Krznarić 2026 The inverse curve shortening flow on the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)5Krznarić 2026 Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick (Klaus Deckelnick and Vanessa Styles, 2026)2Deckelnick 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A novel finite element approximation of anisotropic curve shortening flow

    Klaus Deckelnick and Robert NürnbergWider flows PDF

    Topics Numerics & visualization, Curve shortening, Other curvature flows · Same authors Klaus Deckelnick, Robert Nürnberg · Shared terms “error bound”, “semidiscrete finite”, “numerical simulations”

  2. 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 · Same author Klaus Deckelnick · Shared terms “error bound”, “semidiscrete”, “numerical”

  3. Which shapes can appear in a Curve Shortening Flow Singularity?

    Sigurd Angenent, Evan Patrick Davis, Ellie DeCleene et al.Wider flows PDF

    Topics Numerics & visualization, Curve shortening, Singularities · Shared terms “numerical simulations”

  4. The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature

    Patrick GuidottiWider flows PDF

    Topics Numerics & visualization, Curve shortening, Singularities · Shared terms “numerical”

  5. The inverse curve shortening flow on the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows PDF

    Topics Curve shortening

  6. A curvature flow approach to dorsal closure modelling

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

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

  7. Geometric aspects of the curve shortening flow in the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows PDF

    Topics Curve shortening

  8. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows PDF

    Topics Curve shortening, Singularities

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) 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 usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 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.

  3. 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.