Paper · Nov 13, 2019

On the construction of closed nonconvex nonsoliton ancient mean curvature flows

Theodora Bourni, Mathew Langford, Alexander Mramor

math.DGWider flowsarXiv:1911.05641
View PDF View HTML arXiv page

As on arXiv

Abstract

We construct closed, embedded, ancient mean curvature flows in each dimension with the topology of . These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the self-shrinking doughnuts constructed by Drugan and Nguyen (or, alternatively, Angenent's self shrinking torus when )

View PDF View HTML arXiv page
Submitted
Nov 13, 2019
Subjects
math.DG

Comments: 11 pages. Comments welcome!

Mean curvature flowAncient solutionsSelf-shrinkers & translators

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 Shrinking doughnuts via variational methods (Gregory Drugan and Xuan Hien Nguyen, 2017)5Drugan 2017 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)4Bourni 2019 Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces (Alexander Mramor and Alec Payne, 2019)2Mramor 2019 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)3Bourni 2021 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)1Bourni 2022 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)6Bourni 2022 Convex Ancient Solutions to Anisotropic Curve Shortening Flow (Theodora Bourni and Benjamin Richards, 2023)7Bourni 2023 An Ancient Stacked Pancake Solution to Mean Curvature Flow (Alexander Mramor and Louis Yudowitz, 2025)8Mramor 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

    Topics Ancient solutions, Self-shrinkers & translators · Same authors Theodora Bourni, Alexander Mramor

  2. Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces

    Alexander Mramor and Alec PayneWider flows

    Topics Ancient solutions, Self-shrinkers & translators · Same author Alexander Mramor · Shared terms “nonconvex”, “mean convex”

  3. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics Ancient solutions · Same author Theodora Bourni

  4. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Ancient solutions, Self-shrinkers & translators · Same author Theodora Bourni · Shared terms “Angenent”

  5. Shrinking doughnuts via variational methods

    Gregory Drugan and Xuan Hien NguyenWider flows

    Topics Self-shrinkers & translators · Shared terms “shrinking doughnut”, “shrinking torus”, “self shrinking”

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

    Theodora Bourni and Mat LangfordWider flows

    Topics Ancient solutions · Same author Theodora Bourni

  7. Convex Ancient Solutions to Anisotropic Curve Shortening Flow

    Theodora Bourni and Benjamin RichardsWider flows

    Topics Ancient solutions, Self-shrinkers & translators · Same author Theodora Bourni

  8. An Ancient Stacked Pancake Solution to Mean Curvature Flow

    Alexander Mramor and Louis YudowitzWider flows

    Topics Ancient solutions · Same author Alexander Mramor

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) 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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

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.