Paper · Feb 20, 2019

Ancient gradient flows of elliptic functionals and Morse index

Kyeongsu Choi, Christos Mantoulidis

math.DGWider flowsarXiv:1902.07697v3
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Abstract

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinking equatorial spheres. In the mean curvature flow case in S^3, we classify ancient flows with more relaxed area bounds: they are steady or shrinking equators or Clifford tori. In the embedded curve shortening case in S^2, we completely classify ancient flows of bounded length: they are steady or shrinking circles.

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Submitted
Feb 20, 2019
Last revised
Sep 17, 2020 (v3)
Subjects
math.DG, math.AP
DOI
10.1353/ajm.2022.0010

Comments: Final version; to appear in Amer. J. Math

Mean curvature flowCurve shorteningAncient solutionsHarmonic maps & DeTurck trick

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 Mean curvature flow with generic initial data (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2020)5Chodosh 2020 Classification of ancient flows by sub-affine-critical powers of curvature in ℝ² (Kyeongsu Choi and Liming Sun, 2020)2Choi 2020 Mean curvature flow with generic low-entropy initial data (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2021)8Chodosh 2021 Ancient mean curvature flows with finite total curvature (Kyeongsu Choi, Jiuzhou Huang and Taehun Lee, 2024)4Choi 2024 Uniqueness of tangent flows at infinity for finite-entropy shortening curves (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2024)6Choi 2024 Revisiting generic mean curvature flow in ℝ³ (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2024)1Chodosh 2024 Convergence of Ricci flow and long-time existence of Harmonic map heat flow (Kyeongsu Choi and Yi Lai, 2025)7Choi 2025 Classification of ancient finite-entropy curve shortening flows (Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al., 2026)3Choi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows

    Topics Ancient solutions · Same authors Kyeongsu Choi, Christos Mantoulidis

  2. Classification of ancient flows by sub-affine-critical powers of curvature in

    Kyeongsu Choi and Liming Sun

    Topics Curve shortening, Ancient solutions · Same author Kyeongsu Choi · Shared terms “shrinking circle”

  3. Classification of ancient finite-entropy curve shortening flows

    Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al.Wider flows

    Topics Curve shortening, Ancient solutions · Same author Kyeongsu Choi · Shared terms “shrinking circle”

  4. Ancient mean curvature flows with finite total curvature

    Kyeongsu Choi, Jiuzhou Huang and Taehun LeeWider flows

    Topics Ancient solutions · Same author Kyeongsu Choi · Shared terms “Morse index”

  5. Mean curvature flow with generic initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.

    Topics Ancient solutions · Same authors Kyeongsu Choi, Christos Mantoulidis

  6. Uniqueness of tangent flows at infinity for finite-entropy shortening curves

    Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su et al.Wider flows

    Topics Curve shortening, Ancient solutions · Same author Kyeongsu Choi · Shared terms “shrinking circle”

  7. Convergence of Ricci flow and long-time existence of Harmonic map heat flow

    Kyeongsu Choi and Yi Lai

    Topics Harmonic maps & DeTurck trick, Ancient solutions · Same author Kyeongsu Choi · Shared terms “map heat”, “heat flow”

  8. Mean curvature flow with generic low-entropy initial data

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows

    Topics Mean curvature flow · Same authors Kyeongsu Choi, Christos Mantoulidis

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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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.