Paper · Jan 9, 2019

Nonconvex Surfaces which Flow to Round Points

Alexander Mramor, Alec Payne

math.DGWider flowsarXiv:1901.02863v3
View PDF View HTML arXiv page

As on arXiv

Abstract

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that exist on a uniform time interval, have uniformly bounded diameter, and shrink to round points, yet the sequence of initial surfaces has no subsequence converging in the Gromov-Hausdorff sense. Moreover, we find a sequence of flows which all shrink to round points, yet the initial surfaces converge to a space-filling surface. Also constructed are surfaces of arbitrarily large area which are close in Hausdorff distance to the round sphere yet shrink to round points.

View PDF View HTML arXiv page
Submitted
Jan 9, 2019
Last revised
May 13, 2021 (v3)
Subjects
math.DG, math.AP

Comments: 46 pages, numerous figures. Improved exposition, added some details, and fixed minor errors and typos, as suggested by referees

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 2016, 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 A finiteness theorem via the mean curvature flow with surgery (Alexander Mramor, 2017)2Mramor 2017 Convergence of Riemannian 4-manifolds with L²-curvature bounds (Norman Zergänge, 2017)4Zergänge 2017 Low Entropy and the Mean Curvature Flow with Surgery (Alexander Mramor and Shengwen Wang, 2018)5Mramor 2018 Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces (Alexander Mramor and Alec Payne, 2019)1Mramor 2019 On the construction of closed nonconvex nonsoliton ancient mean curvature flows (Theodora Bourni, Mathew Langford and Alexander Mramor, 2019)3Bourni 2019 Rigidity of the round cylinders in Ricci shrinkers (Yu Li and Bing Wang, 2021)8Li 2021 An Intersection Principle for Mean Curvature Flow (Tang-Kai Lee and Alec Payne, 2025)6Lee 2025 Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres (Reto Buzano and Shengwen Wang, 2026)7Buzano 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces

    Alexander Mramor and Alec PayneWider flows

    Topics Mean curvature flow · Same authors Alexander Mramor, Alec Payne · Shared terms “nonconvex”, “mean convex”, “uniformly bounded”

  2. A finiteness theorem via the mean curvature flow with surgery

    Alexander MramorWider flows

    Topics Limit spaces & RCD · Same author Alexander Mramor · Shared terms “convex hypersurfaces”

  3. On the construction of closed nonconvex nonsoliton ancient mean curvature flows

    Theodora Bourni, Mathew Langford and Alexander MramorWider flows

    Topics Mean curvature flow · Same author Alexander Mramor · Shared terms “nonconvex”, “mean convex”

  4. Convergence of Riemannian -manifolds with -curvature bounds

    Norman ZergängeWider flows

    Topics Limit spaces & RCD · Shared terms “sequence”, “Gromov Hausdorff”, “diameter”

  5. Low Entropy and the Mean Curvature Flow with Surgery

    Alexander Mramor and Shengwen WangWider flows

    Topics Mean curvature flow · Same author Alexander Mramor · Shared terms “mean convex”, “convex hypersurfaces”

  6. An Intersection Principle for Mean Curvature Flow

    Tang-Kai Lee and Alec PayneWider flows

    Topics Mean curvature flow · Same author Alec Payne · Shared terms “Hausdorff”

  7. Perelman's entropy and Gromov-Hausdorff versus dynamical stability of round spheres

    Reto Buzano and Shengwen Wang

    Topics Limit spaces & RCD · Shared terms “round sphere”, “Gromov Hausdorff”, “Hausdorff distance”

  8. Rigidity of the round cylinders in Ricci shrinkers

    Yu Li and Bing Wang

    Topics Limit spaces & RCD · Shared terms “Gromov Hausdorff”

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 in the L² sense. 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.