Paper · Apr 24, 2020

Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder

Beomjun Choi, Kyeongsu Choi, Panagiota Daskalopoulos

math.DGWider flowsarXiv:2004.11754v3
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Abstract

We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time from two opposite ends.

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Submitted
Apr 24, 2020
Last revised
Jul 14, 2022 (v3)
Subjects
math.DG, math.AP

Comments: v2 Mistypes corrected. Lemma 2.6 has been added. The proof of Proposition 3.8 has been simplified v3 Section 6 is revised. Mistypes were corrected

Other curvature flowsGauss curvature flowAncient solutionsSelf-shrinkers & translatorsStability & uniquenessBoundaries & non-compact

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 Uniqueness of closed self-similar solutions to the Gauss curvature flow (Kyeongsu Choi and Panagiota Daskalopoulos, 2016)4Choi 2016 Translating solutions to the Gauss curvature flow with flat sides (Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm Lee, 2016)3Choi 2016 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)6Brendle 2018 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)5Choi 2018 Convergence of Gauss curvature flows to translating solitons (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2019)1Choi 2019 Classification of bubble-sheet ovals in ℝ⁴ (Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al., 2022)7Choi 2022 Rigidity of solitons of the Gauss curvature flow in Euclidean space (Rafael López, 2025)8López 2025 Ancient Gauss curvature flows of bounded width (Beomjun Choi, Kyeongsu Choi and Dongjun Noh, 2025)2Choi 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of Gauss curvature flows to translating solitons

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Gauss curvature flow, Self-shrinkers & translators, Boundaries & non-compact · Same authors Beomjun Choi, Kyeongsu Choi, Panagiota Daskalopoulos · Shared terms “bounded cross”, “cross section”, “translating solitons”

  2. Ancient Gauss curvature flows of bounded width

    Beomjun Choi, Kyeongsu Choi and Dongjun NohWider flows

    Topics Gauss curvature flow, Ancient solutions, Other curvature flows · Same authors Beomjun Choi, Kyeongsu Choi · Shared terms “contained”

  3. Translating solutions to the Gauss curvature flow with flat sides

    Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm LeeWider flows

    Topics Gauss curvature flow, Self-shrinkers & translators, Boundaries & non-compact · Same authors Kyeongsu Choi, Panagiota Daskalopoulos · Shared terms “compact translating”, “convex bounded”, “translating solitons”

  4. Uniqueness of closed self-similar solutions to the Gauss curvature flow

    Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Gauss curvature flow, Other curvature flows, Stability & uniqueness · Same authors Kyeongsu Choi, Panagiota Daskalopoulos

  5. Convergence of Curve Shortening Flow to Translating Soliton

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Same authors Beomjun Choi, Kyeongsu Choi, Panagiota Daskalopoulos · Shared terms “translating solitons”

  6. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

    S. Brendle and K. ChoiWider flows

    Topics Ancient solutions, Self-shrinkers & translators, Boundaries & non-compact · Same author Kyeongsu Choi · Shared terms “translating solitons”

  7. Classification of bubble-sheet ovals in

    Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al.Wider flows

    Topics Ancient solutions · Same authors Beomjun Choi, Panagiota Daskalopoulos

  8. Rigidity of solitons of the Gauss curvature flow in Euclidean space

    Rafael López

    Topics Gauss curvature flow, Self-shrinkers & translators, Other curvature flows · Shared terms “translating solitons”

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) 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.