Paper · Sep 18, 2016

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

Kyeongsu Choi, Panagiota Daskalopoulos

math.DGWider flowsarXiv:1609.05487
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Abstract

We show the uniqueness of strictly convex closed smooth self-similar solutions to the -Gauss curvature flow with . We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the -Gauss curvature flow with shrinks a strictly convex closed smooth hypersurface to a round sphere.

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Submitted
Sep 18, 2016
Subjects
math.DG
Other curvature flowsGauss curvature flowStability & uniqueness

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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 Translating solutions to the Gauss curvature flow with flat sides (Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm Lee, 2016)3Choi 2016 Convergence of Gauss curvature flows to translating solitons (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2019)1Choi 2019 Uniqueness of ancient solutions to Gauss curvature flow asymptotic to a cylinder (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2020)2Choi 2020 A Gauss curvature flow related to the Orlicz-Aleksandrov problem (Bin Chen and Peibiao Zhao, 2021)8Chen 2021 Curvature bound for Lₚ Minkowski problem (Kyeongsu Choi, Minhyun Kim and Taehun Lee, 2023)6Choi 2023 Flow by Gauss curvature to the Minkowski problem of p-harmonic measure (Chao Li and Xia Zhao, 2024)5Li 2024 Regularity of the p-Gauss curvature flow with flat side (Genggeng Huang, Xu-Jia Wang and Yang Zhou, 2024)7Huang 2024 Ancient Gauss curvature flows of bounded width (Beomjun Choi, Kyeongsu Choi and Dongjun Noh, 2025)4Choi 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, Other curvature flows · Same authors Kyeongsu Choi, Panagiota Daskalopoulos

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

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

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

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

    Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm LeeWider flows

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

  4. Ancient Gauss curvature flows of bounded width

    Beomjun Choi, Kyeongsu Choi and Dongjun NohWider flows

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

  5. Flow by Gauss curvature to the Minkowski problem of p-harmonic measure

    Chao Li and Xia ZhaoWider flows

    Topics Gauss curvature flow, Other curvature flows

  6. Curvature bound for Minkowski problem

    Kyeongsu Choi, Minhyun Kim and Taehun LeeWider flows

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

  7. Regularity of the -Gauss curvature flow with flat side

    Genggeng Huang, Xu-Jia Wang and Yang ZhouWider flows

    Topics Gauss curvature flow, Other curvature flows

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

    Bin Chen and Peibiao ZhaoWider flows

    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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)