Paper · Mar 29, 2024

Minimizing movements for the generalized power mean curvature flow

Giovanni Bellettini, Shokhrukh Yu. Kholmatov

math.APWider flowsarXiv:2403.20244v2
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Abstract

Motivated by a conjecture of De Giorgi, we consider the Almgren-Taylor-Wang scheme for mean curvature flow, where the volume penalization is replaced by a term of the form for ranging in a large class of strictly increasing continuous functions. In particular, our analysis covers the case considered by De Giorgi. We show that the generalized minimizing movement scheme converges to the geometric evolution equation where are evolving subsets of is the normal velocity of and is the mean curvature of . We extend our analysis to the anisotropic setting, and in the presence of a driving force. We also show that minimizing movements coincide with the smooth classical solution as long as the latter exists. Finally, we prove that in the absence of forcing, mean convexity and convexity are preserved by the weak flow.

Mean curvature flowWeak & 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 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 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)3Chambolle 2022 Flat flow solution to the mean curvature flow with volume constraint (Vesa Julin, 2023)5Julin 2023 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)6Bellettini 2023 Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in ℝ³ (Shokhrukh Kholmatov, 2024)1Kholmatov 2024 Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions (Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon Almuratov, 2024)7Bellettini 2024 A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain (Tokuhiro Eto and Yoshikazu Giga, 2024)8Eto 2024 Minimizing movements for forced anisotropic curvature flow of droplets (Shokhrukh Yu. Kholmatov, 2024)4Kholmatov 2024 De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach (Tim Laux and Andrea Poiatti, 2026)2Laux 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 minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in

    Shokhrukh KholmatovWider flows PDF

    Topics Weak & singular flows · Same author Shokhrukh Yu. Kholmatov · Shared terms “minimizing movements”, “coincides”

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

    Tim Laux and Andrea PoiattiWider flows PDF

    Topics Weak & singular flows · Shared terms “minimizing movements”, “Giorgi”, “Almgren Taylor”

  3. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Weak & singular flows · Shared terms “minimizing movements”, “scheme converges”, “movement scheme”

  4. Minimizing movements for forced anisotropic curvature flow of droplets

    Shokhrukh Yu. KholmatovWider flows PDF

    Topics Weak & singular flows · Same author Shokhrukh Yu. Kholmatov · Shared terms “minimizing movements”

  5. Flat flow solution to the mean curvature flow with volume constraint

    Vesa JulinWider flows PDF

    Topics Weak & singular flows · Shared terms “minimizing movements”, “volume penalization”, “Almgren Taylor”

  6. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

    Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “classical solution”, “geometric evolution”, “presence”

  7. Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions

    Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon AlmuratovWider flows PDF

    Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “classical solution”

  8. A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

    Tokuhiro Eto and Yoshikazu GigaWider flows PDF

    Topics Weak & singular flows · Shared terms “minimizing movements”, “movement scheme”, “scheme converges”

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) 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.