Paper · Jan 20, 2021

Soliton Solutions to the Curve Shortening Flow on the 2-dimensional hyperbolic plane

Fabio Nunes da Silva, Keti Tenenblat

math.DGWider flowsarXiv:2101.07916v2
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Abstract

We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We use this characterization to provide a qualitative study of the solitons. We show that for each fixed vector there is a 2-parameter family of soliton solutions to the curve shortening flow on the 2-dimensional hyperbolic space. Moreover, we prove that each soliton is defined on the entire real line, it is embedded and its geodesic curvature converges to a constant at each end.

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Submitted
Jan 20, 2021
Last revised
Jan 29, 2021 (v2)
Subjects
math.DG

Comments: 22 pages, in English, 16 figures

Curve shortening3-manifoldsSurfaces

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 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)8Cortissoz 2017 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)7Woolgar 2021 Self-Similar Solutions to the Curvature Flow and its Inverse on the 2-dimensional Light Cone (Fabio Nunes da Silva and Keti Tenenblat, 2021)2da Silva 2021 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)3dos Reis 2022 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)6Fujihara 2024 Geometric aspects of the curve shortening flow in the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)4Krznarić 2026 Solitons for the inverse curve shortening flow on the hyperbolic plane (Ivan Krznarić and Rafael López, 2026)5Krznarić 2026 Soliton Solutions to the Curvature Flow on the 2-dimensional De Sitter Space and Applications (Fábio Nunes da Silva and Edwin Salinas Reyes, 2026)1da Silva 2026
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 Curve shortening, Surfaces, 3-manifolds · Same author Fabio Nunes da Silva · Shared terms “dimensional hyperbolic”, “fixed vector”, “tangent vector”

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

    Fabio Nunes da Silva and Keti TenenblatWider flows

    Topics Surfaces, 3-manifolds · Same authors Fabio Nunes da Silva, Keti Tenenblat · Shared terms “fixed vector”, “tangent vector”, “inner product”

  3. Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows

    Topics Curve shortening, Surfaces · Shared terms “characterization”

  4. Geometric aspects of the curve shortening flow in the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows

    Topics Curve shortening · Shared terms “hyperbolic plane”

  5. Solitons for the inverse curve shortening flow on the hyperbolic plane

    Ivan Krznarić and Rafael LópezWider flows

    Topics Curve shortening · Shared terms “hyperbolic plane”

  6. Curve shortening flows on surfaces that are not convex at infinity

    Naotoshi FujiharaWider flows

    Topics Curve shortening, Surfaces

  7. Self-similar curve shortening flow in hyperbolic 2-space

    Eric Woolgar and Ran XieWider flows

    Topics Curve shortening

  8. On the rate of convergence of the -curve shortening flow

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

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.

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) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

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.

  3. 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.