Paper · Aug 11, 2026

Ancient mean curvature flow asymptotic to a minimal quadratic cone

Junyoung Park

math.DGWider flowsarXiv:2608.11011v4
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Abstract

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an symmetric minimal quadratic cone for , and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a .

Comments: 63 pages, V4 : Journal submission version, fixed minor typos, and extended the main results to more general minimal quadratic cones using same ideas, comments welcome

Mean curvature flowAncient solutionsStability & uniquenessMinimal & CMC surfaces

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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 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)1Han 2023 On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds (Maxwell Stolarski, 2023)3Stolarski 2023 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)2Choi 2024 Existence of 5 minimal tori in 3-spheres of positive Ricci curvature (Adrian Chun-Pong Chu and Yangyang Li, 2024)8Chu 2024 Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions (Junyoung Park, 2025)5Park 2025 When can minimal hypersurfaces be connected by mean curvature flow? (Jingwen Chen and Pedro Gaspar, 2025)7Chen 2025 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)4Bourni 2026 Classification of quadratically pinched translators in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)6Impera 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, Stability & uniqueness · Shared terms “embedded ancient”

  2. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows PDF

    Topics Minimal & CMC surfaces, Ancient solutions

  3. On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

    Maxwell StolarskiWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “Hardt Simon”, “stationary”

  4. 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 “foliation”

  5. Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

    Junyoung ParkWider flows PDF

    Topics Ancient solutions · Same author Junyoung Park

  6. Classification of quadratically pinched translators in higher codimension

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows PDF

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness · Shared terms “quadratic”, “Simon”

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

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Minimal & CMC surfaces

  8. 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

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.