Paper · Feb 18, 2020

Entire downward solitons to the scalar curvature flow in Minkowski space

Pierre Bayard

math.APWider flowsarXiv:2002.07685v2
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As on arXiv

Abstract

Existence and uniqueness in Minkowski space of entire downward translating solitons with prescribed values at infinity for a scalar curvature flow equation. The radial case translates into an ordinary differential equation and the general case into a fully non-linear elliptic PDE on R^n.

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Submitted
Feb 18, 2020
Last revised
Mar 9, 2022 (v2)
Subjects
math.AP

Comments: 26 pages, to appear in Annales de l'Institut Henri Poincaré, Analyse Non Linéaire

Other curvature flowsSelf-shrinkers & translatorsScalar curvatureStability & uniqueness

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 grim hyperplanes for mean curvature flows (Ditter Tasayco and Detang Zhou, 2016)7Tasayco 2016 The prescribed curvature problem for entire hypersurfaces in Minkowski space (Changyu Ren, Zhizhang Wang and Ling Xiao, 2020)1Ren 2020 Entire self-expanders for power of σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)3Wang 2022 Entire convex curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)6Wang 2022 Entire σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)2Wang 2022 On the second boundary value problem for mean curvature flow in Minkowski space (Rongli Huang, 2024)5Huang 2024 Translating solutions and the entire Hessian curvature flow in Minkowski space (Qu Changzheng, Wang Zhizhang and Wo Weifeng, 2024)4Changzheng 2024 Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity (Dake Li, Zhizhang Wang and Shiqi Yin, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The prescribed curvature problem for entire hypersurfaces in Minkowski space

    Changyu Ren, Zhizhang Wang and Ling XiaoWider flows

    Topics Self-shrinkers & translators, Other curvature flows, Stability & uniqueness · Shared terms “downward”, “Minkowski space”, “translating solitons”

  2. Entire curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Self-shrinkers & translators, Other curvature flows · Shared terms “Minkowski space”

  3. Entire self-expanders for power of curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Self-shrinkers & translators, Other curvature flows · Shared terms “Minkowski space”

  4. Translating solutions and the entire Hessian curvature flow in Minkowski space

    Qu Changzheng, Wang Zhizhang and Wo WeifengWider flows

    Topics Self-shrinkers & translators, Other curvature flows · Shared terms “Minkowski space”

  5. On the second boundary value problem for mean curvature flow in Minkowski space

    Rongli HuangWider flows

    Topics Self-shrinkers & translators · Shared terms “Minkowski space”, “translating solitons”

  6. Entire convex curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Self-shrinkers & translators, Other curvature flows · Shared terms “Minkowski space”

  7. Uniqueness of grim hyperplanes for mean curvature flows

    Ditter Tasayco and Detang ZhouWider flows

    Topics Self-shrinkers & translators, Scalar curvature, Stability & uniqueness · Shared terms “translating solitons”

  8. Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity

    Dake Li, Zhizhang Wang and Shiqi YinWider flows

    Topics Self-shrinkers & translators, Other curvature flows · Shared terms “Minkowski space”

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) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.