Paper · May 20, 2021

Translating solutions for a class of quasilinear parabolic initial boundary value problems in Lorentz-Minkowski plane

Ya Gao, Jinghua Li, Jing Mao

math.DGWider flowsarXiv:2105.09482
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Abstract

In this paper, we investigate the evolution of spacelike curves in Lorentz-Minkowski plane along prescribed geometric flows (including the classical curve shortening flow or mean curvature flow as a special case), which correspond to a class of quasilinear parabolic initial boundary value problems, and can prove that this flow exists for all time. Moreover, we can also show that the evolving spacelike curves converge to a spacelike straight line or a spacelike Grim Reaper curve as time tends to infinity.

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Submitted
May 20, 2021
Subjects
math.DG
DOI
10.1063/5.0071167

Comments: 11 pages. Comments are welcome

Mean curvature flowCurve shorteningSelf-shrinkers & translatorsLong-time existence & convergencePhysics

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 Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold Mⁿ×ℝ (Ya Gao, Yi-Juan Gong and Jing Mao, 2020)5Gao 2020 Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)2Gao 2021 An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)3Gao 2021 Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)4Gao 2021 An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane ℝ²₁ (Ya Gao, Chenyang Liu and Jing Mao, 2021)1Gao 2021 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)8Gao 2024 Grim Raindrop: A Translating Solution to Curve Diffusion Flow (W. Jacob Ogden and Micah Warren, 2024)7Ogden 2024 Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces (Baichuan Hu and Xiang Ma, 2026)6Hu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane

    Ya Gao, Chenyang Liu and Jing MaoWider flows

    Topics Physics, Long-time existence & convergence · Same authors Ya Gao, Jing Mao · Shared terms “Minkowski plane”, “Lorentz Minkowski”, “spacelike”

  2. Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows

    Topics Physics, Long-time existence & convergence · Same authors Ya Gao, Jing Mao · Shared terms “Lorentz Minkowski”, “spacelike”, “time tends”

  3. An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows

    Topics Physics, Long-time existence & convergence · Same authors Ya Gao, Jing Mao · Shared terms “Lorentz Minkowski”, “spacelike”, “time tends”

  4. Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows

    Topics Physics, Long-time existence & convergence · Same authors Ya Gao, Jing Mao · Shared terms “Lorentz Minkowski”, “spacelike”, “time tends”

  5. Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold

    Ya Gao, Yi-Juan Gong and Jing MaoWider flows

    Topics Self-shrinkers & translators · Same authors Ya Gao, Jing Mao · Shared terms “translating solutions”

  6. Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces

    Baichuan Hu and Xiang MaWider flows

    Topics Curve shortening, Physics · Shared terms “spacelike curves”

  7. Grim Raindrop: A Translating Solution to Curve Diffusion Flow

    W. Jacob Ogden and Micah WarrenWider flows

    Topics Curve shortening, Self-shrinkers & translators · Shared terms “translating solutions”

  8. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows

    Topics Long-time existence & convergence · Same authors Ya Gao, Jing Mao

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  2. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.