Paper · Aug 14, 2018

Volume preserving mean curvature flow for star-shaped sets

Inwon Kim, Dohyun Kwon

math.APWider flowsarXiv:1808.04922
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Abstract

We study the evolution of star-shaped sets in volume preserving mean curvature flow. Constructed by approximate minimizing movements, our solutions preserve a strong version of star-shapedness. We also show that the solutions converges to a ball as time goes to infinity. For asymptotic behavior of the solutions we use the gradient flow structure of the problem, whereas a modified notion of viscosity solutions is introduced to study the geometric properties of the flow by moving planes method.

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Submitted
Aug 14, 2018
Subjects
math.AP
Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows

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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 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 Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2020)2Julin 2020 On volume-preserving crystalline mean curvature flow (Inwon Kim, Dohyun Kwon and Norbert Požár, 2020)1Kim 2020 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)3Laux 2022 Consistency of the flat flow solution to the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2022)4Julin 2022 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)5Bonforte 2024 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)8Chiesa 2025 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)6Arya 2026 De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach (Tim Laux and Andrea Poiatti, 2026)7Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On volume-preserving crystalline mean curvature flow

    Inwon Kim, Dohyun Kwon and Norbert PožárWider flows

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

  2. Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “asymptotic behavior”, “minimizing movements”

  3. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “gradient flow”

  4. Consistency of the flat flow solution to the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “minimizing movements”

  5. Asymptotic behavior of a diffused interface volume-preserving mean curvature flow

    Matteo Bonforte, Francesco Maggi and Daniel RestrepoWider flows

    Topics Volume-preserving & constrained flows · Shared terms “asymptotic behavior”

  6. Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “minimizing movements”

  7. De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

    Tim Laux and Andrea PoiattiWider flows

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “minimizing movements”

  8. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows

    Topics Volume-preserving & constrained flows, Weak & singular flows

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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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.