Paper · Jul 19, 2021

Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere

Jingwen Chen, Pedro Gaspar

math.DGWider flowsarXiv:2107.09140v2
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Abstract

In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen, and Tonegawa, and it uses the classification of ancient gradient flows in spheres, by K. Choi and C. Mantoulidis, as well as the rigidity of stationary solutions with low Morse index proved by F. Hiesmayr.

Comments: 25 pages. Comments are welcome!

Mean curvature flowAncient solutionsWeak & singular flowsStability & uniquenessPhase fields & multiphase 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 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)6Hensel 2021 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)7Fischer 2022 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)3Steinke 2023 Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow (Likhit Ganedi, Alice Marveggio and Kerrek Stinson, 2024)4Ganedi 2024 Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen–Cahn (J. M. Daniels-Holgate, 2025)2Daniels-Holg 2025 Parabolic free boundary phase transition and mean curvature flow (Jingeon An-Lacroix and Kiichi Tashiro, 2025)8An-Lacroix 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)5Laux 2026 Allen–Cahn and mean curvature flows from Clifford hypersurfaces to equators (Jingwen Chen, 2026)1Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Allen–Cahn and mean curvature flows from Clifford hypersurfaces to equators

    Jingwen ChenWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Jingwen Chen · Shared terms “Allen”, “Cahn”, “equatorial spheres”

  2. Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen–Cahn

    J. M. Daniels-HolgateWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Shared terms “parabolic Allen”, “Brakke motions”, “Cahn”

  3. Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow

    Pascal SteinkeWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Cahn equation”, “Allen Cahn”

  4. Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

    Likhit Ganedi, Alice Marveggio and Kerrek StinsonWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Shared terms “Allen Cahn”, “Cahn equation”

  5. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Allen Cahn”, “Cahn equation”, “Brakke”

  6. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Shared terms “Allen Cahn”, “Cahn equation”, “Brakke”

  7. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

    Julian Fischer and Alice MarveggioWider flows PDF

    Topics Phase fields & multiphase flows, Stability & uniqueness · Shared terms “Allen Cahn”, “Cahn equation”, “gradient flow”

  8. Parabolic free boundary phase transition and mean curvature flow

    Jingeon An-Lacroix and Kiichi TashiroWider flows PDF

    Topics Phase fields & multiphase flows · Shared terms “parabolic Allen”, “Cahn equation”, “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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) 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)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.