Paper · Apr 14, 2022

Classification of convex ancient free boundary mean curvature flows in the ball

Theodora Bourni, Mat Langford

math.DGWider flowsarXiv:2204.06710
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Abstract

We prove that there exists, in every dimension, a unique (modulo rotations about the origin and time translations) convex ancient mean curvature flow in the ball with free boundary on the sphere.

Comments: arXiv admin note: substantial text overlap with arXiv:2111.06051

Mean curvature flowAncient solutionsFree boundary & capillary

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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)1Bourni 2021 Ancient Ricci flows of bounded girth (Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al., 2023)7Bourni 2023 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)6Langford 2023 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)8Bourni 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)3Bourni 2024 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)2Bourni 2026 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)4Bourni 2026 Free boundary flow through cylindrical singularities (Yueheng Bao and Robert Haslhofer, 2026)5Bao 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 curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows PDF

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

  2. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows PDF

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

  3. 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”

  4. 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 · Same authors Theodora Bourni, Mat Langford

  5. Free boundary flow through cylindrical singularities

    Yueheng Bao and Robert HaslhoferWider flows PDF

    Topics Free boundary & capillary, Ancient solutions

  6. A distance comparison principle for curve shortening flow with free boundary

    Mat Langford and Jonathan J. ZhuWider flows PDF

    Topics Free boundary & capillary · Same author Mat Langford

  7. Ancient Ricci flows of bounded girth

    Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al. PDF

    Topics Ancient solutions · Same authors Theodora Bourni, Mat Langford

  8. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows PDF

    Topics Ancient solutions · Same author Theodora Bourni · Shared terms “convex ancient”

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) 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.