Paper · May 28, 2020

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

Vesa Julin, Joonas Niinikoski

math.APWider flowsarXiv:2005.13800
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Abstract

We prove a new quantitative version of the Alexandrov theorem which states that if the mean curvature of a regular set in R^n+1 is close to a constant in L^n-sense, then the set is close to a union of disjoint balls with respect to the Hausdorff distance. This result is more general than the previous quantifications of the Alexandrov theorem and using it we are able to show that in R^2 and R^3 a weak solution of the volume preserving mean curvature flow starting from a set of finite perimeter asymptotically convergences to a disjoint union of equisize balls, up to possible translations. Here by weak solution we mean a flat flow, obtained via the minimizing movements scheme.

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Submitted
May 28, 2020
Subjects
math.AP, math.DG
Journal
Analysis & PDE 16 (2023) 679-710
DOI
10.2140/apde.2023.16.679
Mean curvature flowVolume-preserving & constrained flowsWeak & singular flows

Its neighbourhood

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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 Volume preserving mean curvature flow for star-shaped sets (Inwon Kim and Dohyun Kwon, 2018)4Kim 2018 Volume preserving mean curvature flows near strictly stable sets in flat torus (Joonas Niinikoski, 2019)6Niinikoski 2019 Long time behaviour of discrete volume preserving mean curvature flows (Massimiliano Morini, Marcello Ponsiglione and Emanuele Spadaro, 2020)7Morini 2020 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 Consistency of the flat flow solution to the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2022)1Julin 2022 Flat flow solution to the mean curvature flow with volume constraint (Vesa Julin, 2023)3Julin 2023 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)8Laux 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

    Topics Volume-preserving & constrained flows, Weak & singular flows · Same authors Vesa Julin, Joonas Niinikoski · Shared terms “flat flow”, “regular set”, “movements scheme”

  2. 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 · Same author Vesa Julin · Shared terms “disjoint balls”, “Hausdorff distance”, “flat flow”

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

    Vesa JulinWider flows

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

  4. Volume preserving mean curvature flow for star-shaped sets

    Inwon Kim and Dohyun KwonWider flows

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

  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

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

  6. Volume preserving mean curvature flows near strictly stable sets in flat torus

    Joonas NiinikoskiWider flows

    Topics Volume-preserving & constrained flows · Same author Joonas Niinikoski

  7. Long time behaviour of discrete volume preserving mean curvature flows

    Massimiliano Morini, Marcello Ponsiglione and Emanuele SpadaroWider flows

    Topics Volume-preserving & constrained flows · Shared terms “disjoint balls”, “finite perimeter”

  8. 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”, “movements scheme”

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.