Paper · Dec 27, 2020

On volume-preserving crystalline mean curvature flow

Inwon Kim, Dohyun Kwon, Norbert Požár

math.APWider flowsarXiv:2012.13839
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Abstract

In this work we consider the global existence of volume-preserving crystalline curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address global existence and regularity of the flow for smooth anisotropies. For the non-smooth case we establish global existence results for the types of anisotropies known to be globally well-posed.

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Submitted
Dec 27, 2020
Subjects
math.AP
Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsAnisotropic & nonlocal flowsWeak & singular flowsLong-time existence & convergence

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 2016, 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 level set crystalline mean curvature flow of surfaces (Yoshikazu Giga and Norbert Požár, 2016)3Giga 2016 Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow (Yoshikazu Giga and Norbert Požár, 2017)6Giga 2017 On the self-similar solutions of the crystalline mean curvature flow in three dimensions (Norbert Požár, 2018)7Požár 2018 Volume preserving mean curvature flow for star-shaped sets (Inwon Kim and Dohyun Kwon, 2018)1Kim 2018 Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term (Yoshikazu Giga and Norbert Pozar, 2020)8Giga 2020 Motion by crystalline-like mean curvature: a survey (Yoshikazu Giga and Norbert Požár, 2021)5Giga 2021 Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions (Eric Kim and Dohyun Kwon, 2024)4Kim 2024 Area-preserving crystalline curvature flow in two dimensions (Eric Kim, 2026)2Kim 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume preserving mean curvature flow for star-shaped sets

    Inwon Kim and Dohyun KwonWider flows

    Topics Volume-preserving & constrained flows, Weak & singular flows · Same authors Inwon Kim, Dohyun Kwon · Shared terms “geometric properties”

  2. Area-preserving crystalline curvature flow in two dimensions

    Eric KimWider flows

    Topics Anisotropic & nonlocal flows, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “preserving crystalline”, “crystalline curvature”, “geometric properties”

  3. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”

  4. Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions

    Eric Kim and Dohyun KwonWider flows

    Topics Anisotropic & nonlocal flows, Volume-preserving & constrained flows, Long-time existence & convergence · Same author Dohyun Kwon · Shared terms “Wulff shape”, “reflection”, “preserved”

  5. Motion by crystalline-like mean curvature: a survey

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Norbert Požár · Shared terms “crystalline”

  6. Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”

  7. On the self-similar solutions of the crystalline mean curvature flow in three dimensions

    Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”

  8. Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

    Yoshikazu Giga and Norbert PozarWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”, “crystalline curvature”

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  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.