Paper · Oct 23, 2016

Translating solutions to the Gauss curvature flow with flat sides

Kyeongsu Choi, Panagiota Daskalopoulos, Ki-Ahm Lee

math.DGWider flowsarXiv:1610.07206v2
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Abstract

We derive local estimates for complete non-compact translating solitons of the Gauss curvature flow in which are graphs over a convex domain . This is closely is related to deriving local estimates for the degenerate Monge-Ampére equation. As a result, given a weakly convex bounded domain , we establish the existence of a translating soliton. In particular, when the boundary has a line segment, we show the existence of flat sides of the translator from a local a'priori non-degeneracy estimate near the free-boundary.

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Submitted
Oct 23, 2016
Last revised
Oct 4, 2018 (v2)
Subjects
math.DG
Other curvature flowsGauss curvature flowSelf-shrinkers & translatorsBoundaries & non-compactFree boundary & capillary

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 The scalar curvature flow with a flat side (Hyo Seok Jang and Ki-Ahm Lee, 2018)6Jang 2018 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 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)7Huang 2024 Regularity of the p-Gauss curvature flow with flat side (Genggeng Huang, Xu-Jia Wang and Yang Zhou, 2024)3Huang 2024 Ancient Gauss curvature flows of bounded width (Beomjun Choi, Kyeongsu Choi and Dongjun Noh, 2025)5Choi 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 Kyeongsu Choi, Panagiota Daskalopoulos · Shared terms “translating solitons”

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

  3. 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 · Shared terms “flat side”

  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. 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 · Shared terms “flat side”

  6. The scalar curvature flow with a flat side

    Hyo Seok Jang and Ki-Ahm LeeWider flows

    Topics Free boundary & capillary, Other curvature flows · Same author Ki-Ahm Lee · Shared terms “flat side”, “degenerate”

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

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

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

  8. Gauss curvature flow with shrinking obstacle

    Ki-Ahm Lee and Taehun LeeWider flows

    Topics Gauss curvature flow, Free boundary & capillary, Other curvature flows · Same author Ki-Ahm Lee

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) 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)