Paper · Aug 7, 2019

Construction of High Codimension Ancient Mean Curvature Flows

Douglas Stryker, Ao Sun

math.DGWider flowsarXiv:1908.02688v2
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Abstract

We construct a class of compact ancient solutions to the mean curvature flow in Euclidean space with high codimension. In particular, we construct higher codimensional ancient curve shortening flows. Moreover, we characterize the asymptotic behavior of these solutions. Add on remark: the construction in this paper has been discovered by Altschuler D.-Altschuler S.-Angenent-Wu in [AAAW13].

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Submitted
Aug 7, 2019
Last revised
Sep 5, 2019 (v2)
Subjects
math.DG

Comments: 10 pages, comments are welcomed! After the first version of this preprint was uploaded to arxiv, we discovered that the construction in this paper has been discovered by Altschuler D.-Altschuler S.-Angenent-Wu in [AAAW13]. Since the motivation of our preprint is different from [AAAW13], we still hope the readers could find some innovating ideas from this preprint

Mean curvature flowCurve shorteningAncient solutions

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 Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow (Julius Baldauf and Ao Sun, 2018)7Baldauf 2018 Convex ancient solutions to curve shortening flow (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2019)6Bourni 2019 Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at -∞ (Douglas Stryker and Ao Sun, 2019)1Stryker 2019 Ancient solutions to the Curve Shortening Flow spanning the halfplane (John Man Shun Ma, 2020)4Ma 2020 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)5Bourni 2022 A Compact Non-convex Ancient Curve Shortening Flow (Jumageldi Charyyev, 2022)2Charyyev 2022 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)3Litzinger 2023 On bounds of entropy and total curvature for ancient curve shortening flows (Wei-Bo Su and Kai-Wei Zhao, 2024)8Su 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at

    Douglas Stryker and Ao SunWider flows

    Topics Curve shortening, Ancient solutions · Same authors Douglas Stryker, Ao Sun · Shared terms “compact ancient”

  2. A Compact Non-convex Ancient Curve Shortening Flow

    Jumageldi CharyyevWider flows

    Topics Curve shortening, Ancient solutions

  3. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening · Shared terms “high codimension”, “Altschuler”, “Euclidean”

  4. Ancient solutions to the Curve Shortening Flow spanning the halfplane

    John Man Shun MaWider flows

    Topics Curve shortening, Ancient solutions

  5. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

    Topics Curve shortening, Ancient solutions

  6. Convex ancient solutions to curve shortening flow

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics Curve shortening, Ancient solutions · Shared terms “Angenent”

  7. Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow

    Julius Baldauf and Ao SunWider flows

    Topics Curve shortening · Same author Ao Sun

  8. On bounds of entropy and total curvature for ancient curve shortening flows

    Wei-Bo Su and Kai-Wei ZhaoWider flows

    Topics Curve shortening, Ancient solutions

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) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) 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)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.

  3. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.