Paper · May 5, 2022

Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

Helmut Abels, Felicitas Bürger, Harald Garcke

math.APWider flowsarXiv:2205.02493
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Abstract

We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included.

Comments: 30 pages

Mean curvature flowSingularities

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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 Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion (Helmut Abels, Felicitas Bürger and Harald Garcke, 2022)1Abels 2022 Mean curvature flow with generic low-entropy initial data II (Otis Chodosh, Christos Mantoulidis and Felix Schulze, 2023)5Chodosh 2023 On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces (Richard H Bamler and Bruce Kleiner, 2023)6Bamler 2023 Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow (Helmut Abels, Christoph Hurm and Maximilian Moser, 2024)2Abels 2024 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)7Fujihara 2024 An Intersection Principle for Mean Curvature Flow (Tang-Kai Lee and Alec Payne, 2025)3Lee 2025 The Genus-Decreasing Property of Mean Curvature Flow, I (Brian White, 2026)4White 2026 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)8Sun 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

    Helmut Abels, Felicitas Bürger and Harald GarckeWider flows PDF

    Topics Mean curvature flow · Same authors Helmut Abels, Felicitas Bürger, Harald Garcke · Shared terms “scaled mean”, “diffusion”, “coupling”

  2. Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow

    Helmut Abels, Christoph Hurm and Maximilian MoserWider flows PDF

    Topics Mean curvature flow · Same author Helmut Abels

  3. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows PDF

    Topics Singularities · Shared terms “intersection”

  4. The Genus-Decreasing Property of Mean Curvature Flow, I

    Brian WhiteWider flows PDF

    Topics Singularities · Shared terms “decreasing”

  5. Mean curvature flow with generic low-entropy initial data II

    Otis Chodosh, Christos Mantoulidis and Felix SchulzeWider flows PDF

    Topics Singularities · Shared terms “embedded hypersurfaces”

  6. On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces

    Richard H Bamler and Bruce KleinerWider flows PDF

    Topics Singularities

  7. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows PDF

    Topics Singularities · Shared terms “infinite time”

  8. Mean curvature flows with cylindrical singularities II: stability and genericity

    Ao Sun and Jinxin XueWider flows PDF

    Topics 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.

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) 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)

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.