Paper · May 2, 2024

Ancient mean curvature flows with finite total curvature

Kyeongsu Choi, Jiuzhou Huang, Taehun Lee

math.DGWider flowsarXiv:2405.01062
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Abstract

We construct an -family of ancient graphical mean curvature flows over a minimal hypersurface in of finite total curvature with the Morse index by establishing exponentially fast convergence in terms of . As a corollary, we show that these ancient flows have finite total curvature and finite mass drop. Moreover, one family of these flows is mean convex by a pointwise estimate.

Comments: All comments are welcome

Mean curvature flowAncient solutionsLong-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)1Han 2023 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)7Chen 2024 Revisiting generic mean curvature flow in ℝ³ (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2024)4Chodosh 2024 On bounds of entropy and total curvature for ancient curve shortening flows (Wei-Bo Su and Kai-Wei Zhao, 2024)8Su 2024 When can minimal hypersurfaces be connected by mean curvature flow? (Jingwen Chen and Pedro Gaspar, 2025)6Chen 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)2Bourni 2026 Ancient mean curvature flow asymptotic to a minimal quadratic cone (Junyoung Park, 2026)3Park 2026 Ancient cylindrical flows with fast or slow convergence (Kyeongsu Choi and Robert Haslhofer, 2026)5Choi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows PDF

    Topics Minimal & CMC surfaces, Ancient solutions · Shared terms “finite total”, “total curvature”, “minimal hypersurface”

  2. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows PDF

    Topics Minimal & CMC surfaces, Ancient solutions, Long-time existence & convergence · Shared terms “minimal hypersurface”, “Morse index”, “mean convex”

  3. Ancient mean curvature flow asymptotic to a minimal quadratic cone

    Junyoung ParkWider flows PDF

    Topics Minimal & CMC surfaces, Ancient solutions

  4. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows PDF

    Topics Ancient solutions · Same author Kyeongsu Choi

  5. Ancient cylindrical flows with fast or slow convergence

    Kyeongsu Choi and Robert HaslhoferWider flows PDF

    Topics Ancient solutions · Same author Kyeongsu Choi · Shared terms “mean convex”

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

    Jingwen Chen and Pedro GasparWider flows PDF

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

  7. Mean curvature flow with multiplicity convergence in closed manifolds

    Jingwen Chen and Ao SunWider flows PDF

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

  8. On bounds of entropy and total curvature for ancient curve shortening flows

    Wei-Bo Su and Kai-Wei ZhaoWider flows PDF

    Topics Ancient solutions · Shared terms “total curvature”

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

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

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

  3. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.