Paper · Jan 24, 2023

Flat flow solution to the mean curvature flow with volume constraint

Vesa Julin

math.APWider flowsarXiv:2301.10089
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Abstract

In this paper I will revisit the construction of a global weak solution to the volume preserving mean curvature flow via discrete minimizing movement scheme by Mugnai-Seis-Spadaro (2016). This method is based on the gradient flow approach due to Almgren-Taylor-Wang (1993) and Luckhaus-Strurzenhecker (1995) and my aim is to replace the volume penalization by implementing the volume constraint directly in the discrete scheme, which from practical point of view is perhaps more natural. A technical novelty is the proof of the density estimate which is based on the second variation condition of the energy.

Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows

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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 The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions (Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al., 2021)5Julin 2021 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)7Takasao 2022 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)8Laux 2022 Consistency of the flat flow solution to the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2022)1Julin 2022 Existence of weak solutions to volume-preserving mean curvature flow with obstacles (Jiwoong Jang, 2025)6Jang 2025 Existence of flat flows for volume-preserving mean curvature flow with contact angle (Jiwoong Jang, 2025)4Jang 2025 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)2Arya 2026 De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach (Tim Laux and Andrea Poiatti, 2026)3Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Consistency of the flat flow solution to the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Same author Vesa Julin · Shared terms “flat flow”, “discrete minimizing”, “movement scheme”

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

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Same author Vesa Julin · Shared terms “flat flow”, “discrete minimizing”, “movement scheme”

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

    Tim Laux and Andrea PoiattiWider flows PDF

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

  4. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows PDF

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

  5. The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

    Vesa Julin, Massimiliano Morini, Marcello Ponsiglione et al.Wider flows PDF

    Topics Volume-preserving & constrained flows · Same author Vesa Julin · Shared terms “flat flow”

  6. Existence of weak solutions to volume-preserving mean curvature flow with obstacles

    Jiwoong JangWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “Mugnai Seis”, “Seis Spadaro”

  7. Existence of weak solution to volume preserving mean curvature flow in higher dimensions

    Keisuke TakasaoWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “global weak”, “Mugnai”, “Seis”

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

    Tim LauxWider flows PDF

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

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) 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.