Paper · Oct 11, 2024

Curve shortening flows on surfaces that are not convex at infinity

Naotoshi Fujihara

math.DGWider flowsarXiv:2410.08960
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Abstract

The behavior of the curve shortening flow has been extensively studied. Gage, Hamilton, and Grayson proved that, under the curve shortening flow, an embedded closed curve in the Euclidean plane becomes convex after a finite time and then shrinks to a point while remaining convex. Moreover, Grayson extended these results to surfaces that are convex at infinity and proved results similar to those for plane curves. In this paper, we study the curve shortening flow on surfaces that are not convex at infinity. Specifically, we consider a warped product of a unit circle and an open interval with a strictly increasing warping function. In this setting, we can define a graph property for curves within these warped products. It is known that this graph property is preserved along the curve shortening flow. Similarly to the behavior of the curve shortening flow in the plane, we prove that the curve becomes a graph after a finite time under the curve shortening flow.

Comments: 11 pages, 4 figures

Curve shorteningSingularitiesSurfacesRotational symmetry & warped products

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 2021, 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 Curve shortening flows on rotational surfaces generated by monotone convex functions (Naotoshi Fujihara, 2022)4Fujihara 2022 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)3dos Reis 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)7Peachey 2022 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)1Fujihara 2023 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)6Sobnack 2024 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)2Fujihara 2024 Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds (Naotoshi Fujihara, Toru Kajigaya and Albert Wood, 2026)8Fujihara 2026 Length bounds and rigidity for ancient curve shortening flow on warped product surfaces (Beomseok Goh and Taehun Lee, 2026)5Goh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flows of graphs sliding off to infinity in warped product manifolds

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening, Rotational symmetry & warped products · Same author Naotoshi Fujihara · Shared terms “warping function”

  2. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows PDF

    Topics Rotational symmetry & warped products, Singularities · Same author Naotoshi Fujihara · Shared terms “open interval”, “warping function”, “preserved along”

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

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows PDF

    Topics Curve shortening, Surfaces

  4. Curve shortening flows on rotational surfaces generated by monotone convex functions

    Naotoshi FujiharaWider flows PDF

    Topics Curve shortening · Same author Naotoshi Fujihara

  5. Length bounds and rigidity for ancient curve shortening flow on warped product surfaces

    Beomseok Goh and Taehun LeeWider flows PDF

    Topics Curve shortening, Rotational symmetry & warped products

  6. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows PDF

    Topics Curve shortening · Shared terms “embedded closed”, “closed curves”, “increasing”

  7. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows PDF

    Topics Curve shortening

  8. Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds

    Naotoshi Fujihara, Toru Kajigaya and Albert WoodWider flows PDF

    Topics Curve shortening, Surfaces, Singularities · Same author Naotoshi Fujihara · Shared terms “Grayson”, “finite time”

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.