Paper · Jan 7, 2026

A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions

Giovanni Bellettini, Virginia Lorenzini, Matteo Novaga, Riccardo Scala

math.APWider flowsarXiv:2601.04385
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Abstract

We consider a fourth-order regularization of the curvature flow for an immersed plane curve with fixed boundary, using an elastica-type functional depending on a small positive parameter . We show that the approximating flow smoothly converges, as , to the curvature flow of the curve with Dirichlet boundary conditions for all times before the first singularity of the limit flow.

Other curvature flowsSingularitiesBoundaries & non-compact

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 Type-0 singularities in the network flow – Evolution of trees (Carlo Mantegazza, Matteo Novaga and Alessandra Pluda, 2022)5Mantegazza 2022 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)2Bellettini 2023 Two nonlocal inverse curvature flows of convex closed plane curves (Zezhen Sun, 2023)1Sun 2023 Inverse mean curvature flow with outer obstacle (Kai Xu, 2024)4Xu 2024 On an area-preserving inverse curvature flow for plane curves (Zezhen Sun and Yuting Wu, 2025)6Sun 2025 Well-posedness for the mean curvature flow on the half-space and on bounded domains (Ke Chen, Ruilin Hu and Quoc-Hung Nguyen, 2026)3Chen 2026 Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions (Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al., 2026)8Pu 2026 On global Chern-Calabi flows on compact complex surfaces (Li Ma and Li Sheng, 2026)7Ma 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Two nonlocal inverse curvature flows of convex closed plane curves

    Zezhen SunWider flows PDF

    Topics Other curvature flows, Singularities · Shared terms “plane curves”

  2. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

    Topics Other curvature flows · Same author Giovanni Bellettini

  3. Well-posedness for the mean curvature flow on the half-space and on bounded domains

    Ke Chen, Ruilin Hu and Quoc-Hung NguyenWider flows PDF

    Topics Boundaries & non-compact · Shared terms “Dirichlet boundary”, “boundary condition”, “regularization”

  4. Inverse mean curvature flow with outer obstacle

    Kai XuWider flows PDF

    Topics Boundaries & non-compact, Other curvature flows, Singularities · Shared terms “boundary condition”, “regularization”

  5. Type-0 singularities in the network flow – Evolution of trees

    Carlo Mantegazza, Matteo Novaga and Alessandra PludaWider flows PDF

    Topics Singularities · Same author Matteo Novaga

  6. On an area-preserving inverse curvature flow for plane curves

    Zezhen Sun and Yuting WuWider flows PDF

    Topics Other curvature flows · Shared terms “plane curves”

  7. On global Chern-Calabi flows on compact complex surfaces

    Li Ma and Li ShengWider flows PDF

    Topics Other curvature flows · Shared terms “fourth order”

  8. Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

    Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al.Wider flows PDF

    Topics Boundaries & non-compact, Other curvature flows · Shared terms “boundary condition”

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 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) 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)

In the guidebook

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  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.