Paper · Sep 8, 2021

Self-Similar Solutions to the Curvature Flow and its Inverse on the 2-dimensional Light Cone

Fabio Nunes da Silva, Keti Tenenblat

math.DGWider flowsarXiv:2109.03677
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Abstract

We show that the solutions to the curvature flow (CF) for curves on the 2-dimensional light cone are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperboles are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant from being the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. The curve is a soliton solution when . We prove that, for each vector there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Moreover, at each end of such a curve, the curvature is either unbounded or it tends to or to the constant . Explicitly given soliton solutions are included and some self-similar solutions on the light cone, are visualized.

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Submitted
Sep 8, 2021
Subjects
math.DG

Comments: 10 figures

Other curvature flowsInverse mean curvature flowAncient solutions3-manifoldsSurfacesPhysicsNumerics & visualization

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications

    Fábio Nunes da Silva and Edwin Salinas ReyesWider flows

    Topics Inverse mean curvature flow, Surfaces, Physics · Same author Fabio Nunes da Silva · Shared terms “fixed vector”, “spacelike curves”, “tangent vector”

  2. Geometric flows and space-periodic solitons on the light-cone

    Yun YangWider flows

    Topics 3-manifolds, Other curvature flows · Shared terms “light cone”, “dimensional Minkowski”, “Minkowski space”

  3. On the inverse mean curvature flow by parallel hypersurfaces in space forms

    Alancoc dos Santos Alencar and Keti TenenblatWider flows

    Topics Inverse mean curvature flow, Ancient solutions, Other curvature flows · Same author Keti Tenenblat · Shared terms “explicitly”

  4. Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

    Miroslav Kolar and Daniel SevcovicWider flows

    Topics Numerics & visualization, Surfaces, 3-manifolds

  5. Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space

    Gregório Silva Neto and Vanessa SilvaWider flows

    Topics Inverse mean curvature flow, 3-manifolds, Other curvature flows · Shared terms “Minkowski space”

  6. A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

    Wilhelm Klingenberg, Ben Lambert and Julian ScheuerWider flows

    Topics Physics, 3-manifolds · Shared terms “Minkowski space”, “dimensional Minkowski”, “spacelike”

  7. Inverse curvature flow of closed Legendre curves

    Takashi Kagaya and Masatomo TakahashiWider flows

    Topics Inverse mean curvature flow, Other curvature flows

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

    Baichuan Hu and Xiang MaWider flows

    Topics Physics · Shared terms “light cone”, “spacelike curves”

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) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.