Paper · Nov 22, 2022

Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow

Theodora Bourni, Mat Langford, Stephen Lynch

math.DGWider flowsarXiv:2211.12639
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Abstract

We make rigorous an old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.

Mean curvature flowLimit spaces & RCD

Its neighbourhood

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 Compactness and rigidity of self-shrinking surfaces (Tang-Kai Lee, 2021)6Lee 2021 Convergence of Hermitian manifolds and the Type IIB flow (Nikita Klemyatin, 2021)8Klemyatin 2021 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)1Langford 2021 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)4Bourni 2021 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)2Bourni 2022 Ancient Ricci flows of bounded girth (Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente et al., 2023)3Bourni 2023 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)5Bourni 2026 A note on generalized BV mean curvature flow with a critical forcing term (Kiichi Tashiro, 2026)7Tashiro 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows PDF

    Topics Mean curvature flow · Same authors Mat Langford, Stephen Lynch

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

    Theodora Bourni and Mat LangfordWider flows PDF

    Topics Mean curvature flow · Same authors Theodora Bourni, Mat Langford

  3. Ancient Ricci flows of bounded girth

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

    Same authors Theodora Bourni, Mat Langford · Shared terms “Hamilton”

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

    Theodora Bourni and Mat LangfordWider flows PDF

    Same authors Theodora Bourni, Mat Langford

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

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows PDF

    Same authors Theodora Bourni, Mat Langford

  6. Compactness and rigidity of self-shrinking surfaces

    Tang-Kai LeeWider flows PDF

    Topics Limit spaces & RCD · Shared terms “compactness”

  7. A note on generalized BV mean curvature flow with a critical forcing term

    Kiichi TashiroWider flows PDF

    Topics Limit spaces & RCD · Shared terms “compactness”

  8. Convergence of Hermitian manifolds and the Type IIB flow

    Nikita Klemyatin PDF

    Topics Limit spaces & RCD · Shared terms “Hamilton”, “compactness”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.