Paper · Feb 9, 2026

Ancient solutions to free boundary mean curvature flow

Theodora Bourni, Giada Franz

math.DGWider flowsarXiv:2602.08850
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Abstract

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.

Comments: 27 pages, 1 figure

Mean curvature flowAncient solutionsStability & uniquenessFree boundary & capillaryLong-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 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)5Bourni 2021 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)1Bourni 2022 Free boundary flow with surgery (Robert Haslhofer, 2023)6Haslhofer 2023 Free boundary minimal disks in convex balls (Robert Haslhofer and Daniel Ketover, 2023)3Haslhofer 2023 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)4Han 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)2Bourni 2024 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)7Choi 2024 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)8Bourni 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Classification of convex ancient free boundary mean curvature flows in the ball

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Free boundary & capillary, Ancient solutions · Same author Theodora Bourni · Shared terms “convex ancient”

  2. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows PDF

    Topics Free boundary & capillary, Ancient solutions · Same author Theodora Bourni · Shared terms “convex ancient”

  3. Free boundary minimal disks in convex balls

    Robert Haslhofer and Daniel KetoverWider flows PDF

    Topics Minimal & CMC surfaces, Free boundary & capillary · Shared terms “convex foliation”, “smooth free”, “mean convex”

  4. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows PDF

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

  5. Classification of convex ancient free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Free boundary & capillary, Ancient solutions · Same author Theodora Bourni · Shared terms “convex ancient”

  6. Free boundary flow with surgery

    Robert HaslhoferWider flows PDF

    Topics Minimal & CMC surfaces, Free boundary & capillary, Long-time existence & convergence · Shared terms “mean convex”

  7. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows PDF

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

  8. Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows PDF

    Topics Free boundary & capillary, Stability & uniqueness · Same author Theodora Bourni

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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

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.