Paper · Jan 4, 2019

Convergence of Gauss curvature flows to translating solitons

Beomjun Choi, Kyeongsu Choi, Panagiota Daskalopoulos

math.DGWider flowsarXiv:1901.01080v4
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Abstract

We address the asymptotic behavior of the -Gauss curvature flow, for , with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow converges, as , locally smoothly to a translating soliton which is uniquely determined by the asymptotic cylinder of the initial hypersurface.

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Submitted
Jan 4, 2019
Last revised
Jan 12, 2022 (v4)
Subjects
math.DG, math.AP

Comments: To be published in Adv. Math

Other curvature flowsGauss curvature flowSelf-shrinkers & translatorsBoundaries & non-compact

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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 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)2Choi 2016 Convergence of Curve Shortening Flow to Translating Soliton (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2018)5Choi 2018 Evolution of non-compact hypersurfaces by inverse mean curvature (Beomjun Choi and Panagiota Daskalopoulos, 2018)7Choi 2018 Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2020)1Choi 2020 Rigidity of solitons of the Gauss curvature flow in Euclidean space (Rafael López, 2025)6López 2025 Invariant λ-translators for the Gauss curvature flow in Euclidean space (Muhittin Evren Aydin and Rafael López, 2025)8Aydin 2025 Ancient Gauss curvature flows of bounded width (Beomjun Choi, Kyeongsu Choi and Dongjun Noh, 2025)3Choi 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder

    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. 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 “translating solitons”

  3. Ancient Gauss curvature flows of bounded width

    Beomjun Choi, Kyeongsu Choi and Dongjun NohWider flows

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

  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 · 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”, “compact convex”, “asymptotic behavior”

  6. 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”

  7. Evolution of non-compact hypersurfaces by inverse mean curvature

    Beomjun Choi and Panagiota DaskalopoulosWider flows

    Topics Boundaries & non-compact, Other curvature flows · Same authors Beomjun Choi, Panagiota Daskalopoulos · Shared terms “compact convex”, “initial hypersurface”, “convex hypersurfaces”

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

    Muhittin Evren Aydin and Rafael LópezWider flows

    Topics Gauss curvature flow, Self-shrinkers & translators, Other curvature flows

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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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)