Paper · Jun 16, 2021

Continuous dependence of curvature flow on initial conditions

Michael Gene Dobbins

math.DGWider flowsarXiv:2106.08907
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Abstract

We study the evolution of a Jordan curve on the 2-sphere by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the evolution of the curve depends continuously on the initial curve in Fréchet distance in the case where the curve bisects the sphere. This even holds in the limit as time goes to infinity. This builds on Joseph Lauer's work on existence and uniqueness of solutions to the curvature flow problem on the sphere when the initial curve is not smooth.

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Submitted
Jun 16, 2021
Subjects
math.DG, math.AP, math.MG
Curve shorteningOther curvature flowsWeak & singular flowsStability & 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 The evolution of Jordan curves on S² by curve shortening flow (Joseph Lauer, 2016)2Lauer 2016 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)5Cortissoz 2017 From conservative to dissipative systemsthrough quadratic change of time, with application to the curve-shortening flow (Yann Brenier and Xianglong Duan, 2017)4Brenier 2017 Self-similar curve shortening flow in hyperbolic 2-space (Eric Woolgar and Ran Xie, 2021)7Woolgar 2021 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)3Peachey 2022 Singularities of low entropy high codimension curve shortening flow (Florian Litzinger, 2023)8Litzinger 2023 The continuity of curvature flow in Fréchet distance (Shiyi Ma, 2023)1Ma 2023 Singularities of Curve Shortening Flow with Convex Projections (Qi Sun, 2025)6Sun 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The continuity of curvature flow in Fréchet distance

    Shiyi MaWider flows

    Topics Curve shortening, Weak & singular flows, Other curvature flows · Shared terms “Fréchet distance”, “curve depends”, “Jordan curve”

  2. The evolution of Jordan curves on by curve shortening flow

    Joseph LauerWider flows

    Topics Curve shortening, Weak & singular flows, Stability & uniqueness · Shared terms “Jordan curve”, “set flow”, “level set”

  3. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening, Stability & uniqueness

  4. From conservative to dissipative systemsthrough quadratic change of time, with application to the curve-shortening flow

    Yann Brenier and Xianglong DuanWider flows

    Topics Curve shortening · Shared terms “initial conditions”

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

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Topics Curve shortening

  6. Singularities of Curve Shortening Flow with Convex Projections

    Qi SunWider flows

    Topics Curve shortening, Stability & uniqueness

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

    Eric Woolgar and Ran XieWider flows

    Topics Curve shortening

  8. Singularities of low entropy high codimension curve shortening flow

    Florian LitzingerWider flows

    Topics Curve shortening · Shared terms “initial curve”

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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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. 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.