Paper · Nov 26, 2023

Ancient mean curvature flows from minimal hypersurfaces

Yongheng Han

math.DGWider flowsarXiv:2311.15278v4Geom. Dedicata, 220.15, (2026)
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Abstract

For , we construct -dimensional family of embedded ancient solutions to mean curvature flow arise from an unstable minimal hypersurface with finite total curvature in , where is the Morse index of the Jacobi operator on .

Comments: Final version, to appear in a journal

Mean curvature flowAncient solutionsStability & uniquenessMinimal & 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 Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders (Katherine Castro and César Rosales, 2022)7Castro 2022 Mean curvature flow with multiplicity 2 convergence in closed manifolds (Jingwen Chen and Ao Sun, 2024)6Chen 2024 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)1Choi 2024 Singularities of mean curvature flow with bounded mean curvature and Morse index (Yongheng Han, 2025)5Han 2025 Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons (Yukai Sun and Guangrui Zhu, 2025)8Sun 2025 When can minimal hypersurfaces be connected by mean curvature flow? (Jingwen Chen and Pedro Gaspar, 2025)4Chen 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)3Bourni 2026 Ancient mean curvature flow asymptotic to a minimal quadratic cone (Junyoung Park, 2026)2Park 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 with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows PDF

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

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

    Junyoung ParkWider flows PDF

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness · Shared terms “embedded ancient”

  3. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows PDF

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness · Shared terms “minimal hypersurface”, “Morse index”, “unstable”

  4. 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”

  5. Singularities of mean curvature flow with bounded mean curvature and Morse index

    Yongheng HanWider flows PDF

    Topics Mean curvature flow · Same author Yongheng Han · Shared terms “Morse index”

  6. Mean curvature flow with multiplicity convergence in closed manifolds

    Jingwen Chen and Ao SunWider flows PDF

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

  7. Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

    Katherine Castro and César RosalesWider flows PDF

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

  8. Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

    Yukai Sun and Guangrui Zhu PDF

    Topics Minimal & CMC surfaces · 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) 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) 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) 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.