Paper · Nov 27, 2020

Stationary sets of the mean curvature flow with a forcing term

Vesa Julin, Joonas Niinikoski

math.APWider flowsarXiv:2011.13666v3
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Abstract

We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space of dimension at least 2. Our main results states that tangential balls in under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in for a constant forcing term as finite unions of equisized balls with mutually positive distance.

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Submitted
Nov 27, 2020
Last revised
Jun 24, 2021 (v3)
Subjects
math.AP

Comments: Minor corrections and two new reference added

Mean curvature flow

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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 Existence of weak solution for mean curvature flow with transport term and forcing term (Keisuke Takasao, 2019)7Takasao 2019 Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane (Nicola Fusco, Vesa Julin and Massimiliano Morini, 2020)1Fusco 2020 Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2020)2Julin 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)3Julin 2022 Flat flow solution to the mean curvature flow with volume constraint (Vesa Julin, 2023)6Julin 2023 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)4Arya 2026 A note on generalized BV mean curvature flow with a critical forcing term (Kiichi Tashiro, 2026)8Tashiro 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane

    Nicola Fusco, Vesa Julin and Massimiliano MoriniWider flows

    Topics Mean curvature flow · Same author Vesa Julin · Shared terms “stationary sets”, “forcing term”, “constant forcing”

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

    Vesa Julin and Joonas NiinikoskiWider flows

    Topics Mean curvature flow · Same authors Vesa Julin, Joonas Niinikoski · Shared terms “flat flow”

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

    Vesa Julin and Joonas NiinikoskiWider flows

    Topics Mean curvature flow · Same authors Vesa Julin, Joonas Niinikoski · Shared terms “flat flow”

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

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows

    Topics Mean curvature flow · Same author Vesa Julin · Shared terms “flat flow”, “Morini”

  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 Mean curvature flow · Same author Vesa Julin · Shared terms “flat flow”, “finite union”

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

    Vesa JulinWider flows

    Topics Mean curvature flow · Same author Vesa Julin · Shared terms “flat flow”

  7. Existence of weak solution for mean curvature flow with transport term and forcing term

    Keisuke TakasaoWider flows

    Topics Mean curvature flow · Shared terms “forcing term”

  8. A note on generalized BV mean curvature flow with a critical forcing term

    Kiichi TashiroWider flows

    Topics Mean curvature flow · Shared terms “forcing term”

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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)

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.