Paper · Jul 17, 2025

Rigidity of solitons of the Gauss curvature flow in Euclidean space

Rafael López

math.DGarXiv:2507.12863
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Abstract

In this paper, we consider -translating solitons and -shrinkers of the Gauss curvature flow in Euclidean space. We prove that planes and circular cylinders are the only -translating solitons with constant mean curvature. We also prove that planes, spheres and circular cylinders are the only -shrinkers with constant mean curvature. We give a classification of the -translating solitons and -shrinkers with one constant principal curvature.

Other curvature flowsGauss curvature flowRicci solitonsSelf-shrinkers & translatorsStability & 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 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)7Chen 2021 Translators of the Gauss curvature flow (Muhittin Evren Aydin and Rafael López, 2022)2Aydin 2022 The chord Gauss curvature flow and its Lₚ chord Minkowski problem (Jinrong Hu, Yong Huang, Jian Lu et al., 2023)6Hu 2023 Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity (Xia Zhao and Peibiao Zhao, 2024)8Zhao 2024 Regularity of the p-Gauss curvature flow with flat side (Genggeng Huang, Xu-Jia Wang and Yang Zhou, 2024)5Huang 2024 Invariant λ-translators for the Gauss curvature flow in Euclidean space (Muhittin Evren Aydin and Rafael López, 2025)1Aydin 2025 Gauss curvature solitons on invariant surfaces in the homogeneous space Sol (Rafael Belli and Rafael López, 2026)4Belli 2026 Classification of invariant Gauss curvature solitons in the Heisenberg space nil (Rafael Belli and Rafael López, 2026)3Belli 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Invariant -translators for the Gauss curvature flow in Euclidean space

    Muhittin Evren Aydin and Rafael LópezWider flows PDF

    Topics Gauss curvature flow, Self-shrinkers & translators, Other curvature flows · Same author Rafael López · Shared terms “Euclidean space”

  2. Translators of the Gauss curvature flow

    Muhittin Evren Aydin and Rafael LópezWider flows PDF

    Topics Gauss curvature flow, Self-shrinkers & translators, Other curvature flows · Same author Rafael López · Shared terms “Euclidean space”

  3. Classification of invariant Gauss curvature solitons in the Heisenberg space

    Rafael Belli and Rafael LópezWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Same author Rafael López

  4. Gauss curvature solitons on invariant surfaces in the homogeneous space Sol

    Rafael Belli and Rafael LópezWider flows PDF

    Topics Gauss curvature flow, Other curvature flows, Stability & uniqueness · Same author Rafael López

  5. 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

  6. 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

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

    Bin Chen and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Other curvature flows

  8. Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity

    Xia Zhao and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Other curvature flows, Stability & uniqueness

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 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.