Paper · Jul 8, 2019

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

Joonas Niinikoski

math.APWider flowsarXiv:1907.03618v4
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Abstract

In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus in low dimensions . The result says roughly that if the initial set is near to a strictly stable set in in -sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in -sense.

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Submitted
Jul 8, 2019
Last revised
Jun 27, 2021 (v4)
Subjects
math.AP

Comments: Several typos corrected

Mean curvature flowVolume-preserving & constrained flowsStability & uniquenessLong-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 Long time behaviour of discrete volume preserving mean curvature flows (Massimiliano Morini, Marcello Ponsiglione and Emanuele Spadaro, 2020)6Morini 2020 Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2020)5Julin 2020 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)3Huang 2021 Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus (Daniele De Gennaro and Anna Kubin, 2022)2De Gennaro 2022 Consistency of the flat flow solution to the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2022)4Julin 2022 Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus (Daniele De Gennaro, Antonia Diana, Andrea Kubin et al., 2023)1De Gennaro 2023 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)7Bonforte 2024 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)8Arya 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

    Daniele De Gennaro, Antonia Diana, Andrea Kubin et al.Wider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Stability & uniqueness · Shared terms “flat torus”, “strictly stable”, “translate”

  2. Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torus

    Daniele De Gennaro and Anna KubinWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “flat torus”, “strictly stable”, “initial set”

  3. Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence, Stability & uniqueness · Shared terms “converges exponentially”

  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 · Same author Joonas Niinikoski · Shared terms “initial set”

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

    Vesa Julin and Joonas NiinikoskiWider flows

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

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

    Massimiliano Morini, Marcello Ponsiglione and Emanuele SpadaroWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “initial set”

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

    Matteo Bonforte, Francesco Maggi and Daniel RestrepoWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence

  8. 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, Long-time existence & convergence · Shared terms “exponentially fast”, “converges exponentially”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 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.