Paper · Sep 11, 2022

Classification of bubble-sheet ovals in

Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du, Robert Haslhofer, Natasa Sesum

math.DGWider flowsarXiv:2209.04931v3Geom. Topol. 29 (2025) 931-1016
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Abstract

In this paper, we prove that any bubble-sheet oval for the mean curvature flow in , up to scaling and rigid motion, either is the -symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of -symmetric ancient ovals constructed by the third and fourth author. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.

Comments: 88 pages. In v2, we added details to the proof of Corollary 1.5. In v3, minor corrections are made following referee's suggestion. The paper will appear in Geom. Topol

Mean curvature flowAncient solutionsHomogeneous spaces

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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 Hearing the shape of ancient noncollapsed flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2021)1Du 2021 A nonexistence result for rotating mean curvature flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2022)7Du 2022 Classification of ancient noncollapsed flows in ℝ⁴ (Kyeongsu Choi and Robert Haslhofer, 2024)5Choi 2024 Rigidity of ancient ovals in higher dimensional mean curvature flow (Beomjun Choi, Wenkui Du and Jingze Zhu, 2025)3Choi 2025 Mean curvature flow near a peanut solution (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2025)6Angenent 2025 Classification of ancient ovals in higher dimensional mean curvature flow (Beomjun Choi, Wenkui Du and Ziyi Zhao, 2026)2Choi 2026 Unique asymptotics of SO(k)× SO(n-k+1) symmetric ancient ovals of Ricci flow (Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al., 2026)4Daskalopoulo 2026 Ancient cylindrical flows with fast or slow convergence (Kyeongsu Choi and Robert Haslhofer, 2026)8Choi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Hearing the shape of ancient noncollapsed flows in

    Wenkui Du and Robert HaslhoferWider flows PDF

    Topics Ancient solutions · Same authors Wenkui Du, Robert Haslhofer · Shared terms “bubble sheet”, “ovals constructed”, “symmetric ancient”

  2. Classification of ancient ovals in higher dimensional mean curvature flow

    Beomjun Choi, Wenkui Du and Ziyi ZhaoWider flows PDF

    Topics Ancient solutions · Same authors Beomjun Choi, Wenkui Du · Shared terms “ancient ovals”, “ovals constructed”, “rigid motion”

  3. Rigidity of ancient ovals in higher dimensional mean curvature flow

    Beomjun Choi, Wenkui Du and Jingze ZhuWider flows PDF

    Topics Ancient solutions · Same authors Beomjun Choi, Wenkui Du · Shared terms “ancient ovals”, “ovals constructed”, “symmetric ancient”

  4. Unique asymptotics of symmetric ancient ovals of Ricci flow

    Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al. PDF

    Topics Ancient solutions · Same authors Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum · Shared terms “ancient ovals”, “symmetric ancient”, “instance”

  5. Classification of ancient noncollapsed flows in

    Kyeongsu Choi and Robert HaslhoferWider flows PDF

    Topics Ancient solutions, Homogeneous spaces · Same author Robert Haslhofer · Shared terms “bubble sheet”, “ovals constructed”, “symmetric ancient”

  6. Mean curvature flow near a peanut solution

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows PDF

    Topics Ancient solutions · Same authors Panagiota Daskalopoulos, Natasa Sesum · Shared terms “ancient ovals”

  7. A nonexistence result for rotating mean curvature flows in

    Wenkui Du and Robert HaslhoferWider flows PDF

    Topics Ancient solutions · Same authors Wenkui Du, Robert Haslhofer

  8. Ancient cylindrical flows with fast or slow convergence

    Kyeongsu Choi and Robert HaslhoferWider flows PDF

    Topics Ancient solutions · Same author Robert Haslhofer · Shared terms “ancient ovals”, “Hershkovits”

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) 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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

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.