Paper · Oct 7, 2025

Ancient Gauss Curvature Flows of Bounded Width

Beomjun Choi, Kyeongsu Choi, Dongjun Noh

math.DGWider flowsarXiv:2510.05614
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Abstract

In this paper, we construct a pancake-like ancient compact solution with flat sides to the Gauss curvature flow, contained in a slab. Also, we construct sausage-like ancient compact solutions to the -Gauss curvature flow with , asymptotic to a round cylinder.

Comments: 26 pages, 4 figures

Other curvature flowsGauss curvature flowAncient solutions

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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 A Gauss curvature flow related to the Orlicz-Aleksandrov problem (Bin Chen and Peibiao Zhao, 2021)5Chen 2021 Curvature bound for Lₚ Minkowski problem (Kyeongsu Choi, Minhyun Kim and Taehun Lee, 2023)3Choi 2023 The chord Gauss curvature flow and its Lₚ chord Minkowski problem (Jinrong Hu, Yong Huang, Jian Lu et al., 2023)4Hu 2023 Gauss curvature flow with shrinking obstacle (Ki-Ahm Lee and Taehun Lee, 2023)8Lee 2023 Long time regularity of the p-Gauss curvature flow with flat side (G. Huang, X. -J. Wang and Y. Zhou, 2024)2Huang 2024 Gauss curvature flow to the Lₚ-Gaussian chord Minkowski problem (Xia Zhao and Peibiao Zhao, 2024)7Zhao 2024 Regularity of the p-Gauss curvature flow with flat side (Genggeng Huang, Xu-Jia Wang and Yang Zhou, 2024)1Huang 2024 Invariant λ-translators for the Gauss curvature flow in Euclidean space (Muhittin Evren Aydin and Rafael López, 2025)6Aydin 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Regularity of the -Gauss curvature flow with flat side

    Genggeng Huang, Xu-Jia Wang and Yang ZhouWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “flat side”

  2. Long time regularity of the -Gauss curvature flow with flat side

    G. Huang, X. -J. Wang and Y. ZhouWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “flat side”

  3. Curvature bound for Minkowski problem

    Kyeongsu Choi, Minhyun Kim and Taehun LeeWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Same author Kyeongsu Choi

  4. The chord Gauss curvature flow and its chord Minkowski problem

    Jinrong Hu, Yong Huang, Jian Lu et al.Wider flows PDF

    Topics Gauss curvature flow, Other curvature flows

  5. A Gauss curvature flow related to the Orlicz-Aleksandrov problem

    Bin Chen and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Other curvature flows

  6. Invariant -translators for the Gauss curvature flow in Euclidean space

    Muhittin Evren Aydin and Rafael LópezWider flows PDF

    Topics Gauss curvature flow, Other curvature flows

  7. Gauss curvature flow to the -Gaussian chord Minkowski problem

    Xia Zhao and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Other curvature flows

  8. Gauss curvature flow with shrinking obstacle

    Ki-Ahm Lee and Taehun LeeWider flows PDF

    Topics Gauss curvature flow, Other curvature flows

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… Start with: Shapes of worn stones (Firey, 1974) 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. 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.