Paper · May 19, 2025

Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

Miroslav Kolar, Daniel Sevcovic

math.APWider flowsarXiv:2505.12775v2
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As on arXiv

Abstract

We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of nonlinear parabolic equations describing the motion of curves belonging to a given two-dimensional manifold. Using the abstract theory of analytic semiflows, we prove the local existence, uniqueness of Hölder smooth solutions to the governing system of nonlinear parabolic equations for the position vector parametrization of evolving curves. We apply the method of flowing finite volumes in combination with the methods of lines for numerical approximation of the governing equations. Qualitative analytical results are illustrated by various numerical experiments.

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Submitted
May 19, 2025
Last revised
Dec 22, 2025 (v2)
Subjects
math.AP
Mean curvature flow3-manifoldsSurfacesStability & uniquenessNumerics & visualization

Its neighbourhood

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature

    Patrick GuidottiWider flows

    Topics Numerics & visualization · Shared terms “numerical experiments”, “dimensional Euclidean”, “qualitative”

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

    Roman BahadursinghWider flows

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

  3. Self-Similar Solutions to the Curvature Flow and its Inverse on the 2-dimensional Light Cone

    Fabio Nunes da Silva and Keti TenenblatWider flows

    Topics Numerics & visualization, Surfaces, 3-manifolds

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

    Klaus Deckelnick and Vanessa StylesWider flows

    Topics Numerics & visualization · Shared terms “numerical experiments”, “position vector”, “approximation”

  5. A curvature flow approach to dorsal closure modelling

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

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

  6. Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles

    Marco Cicalese and Andrea KubinWider flows

    Topics Surfaces, Stability & uniqueness · Shared terms “governing system”

  7. Stability of piecewise flat Ricci flow

    Rory Conboye

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

  8. Self-shrinkers whose asymptotic cones fatten

    Daniel KetoverWider flows

    Topics Numerics & visualization · Shared terms “numerical experiments”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.