Paper · Feb 7, 2024

Mean curvature flow with multiplicity convergence in closed manifolds

Jingwen Chen, Ao Sun

math.DGWider flowsarXiv:2402.04521v2
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Abstract

We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity as time approaches infinity.

Comments: 19 pages, 3 figures. Revisions have been made based on the comments of an anonymous referee. To appear in Mathematische Annalen

Mean curvature flowLong-time existence & convergenceMinimal & CMC surfaces

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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 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 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)5Han 2023 Mean curvature flow with multiplicity 2 convergence in ℝ³ (Jingwen Chen and Ao Sun, 2023)1Chen 2023 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)6Choi 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)7Chu 2024 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)2Mramor 2024 When can minimal hypersurfaces be connected by mean curvature flow? (Jingwen Chen and Pedro Gaspar, 2025)3Chen 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)8Bourni 2026 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)4Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow with multiplicity 2 convergence in

    Jingwen Chen and Ao SunWider flows PDF

    Topics Long-time existence & convergence · Same authors Jingwen Chen, Ao Sun · Shared terms “multiplicity”, “embedded connected”, “immortal mean”

  2. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces · Same author Ao Sun · Shared terms “closed manifolds”, “multiplicity”, “smooth embedded”

  3. When can minimal hypersurfaces be connected by mean curvature flow?

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Minimal & CMC surfaces · Same author Jingwen Chen · Shared terms “minimal hypersurface”

  4. Closed mean curvature flows with prescribed tangent flows

    Jingwen Chen, Tang-Kai Lee, Ao Sun et al. PDF

    Topics Mean curvature flow · Same authors Jingwen Chen, Ao Sun

  5. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “minimal hypersurface”

  6. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows PDF

    Topics Minimal & CMC surfaces, Long-time existence & convergence · Shared terms “minimal hypersurface”

  7. Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

    Adrian Chun-Pong Chu and Yangyang LiWider flows PDF

    Topics Minimal & CMC surfaces

  8. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows PDF

    Topics Minimal & CMC surfaces, Long-time existence & convergence · Shared terms “minimal hypersurface”

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) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.