Paper · Sep 18, 2023

Existence and Morse Index of two free boundary embedded geodesics on Riemannian 2-disks with convex boundary

Dongyeong Ko

math.DGWider flowsarXiv:2309.09896v2
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Abstract

We prove that a free boundary curve shortening flow on closed surfaces with a strictly convex boundary remains noncollapsed for a finite time in the sense of the reflected chord-arc profile introduced by Langford-Zhu. This shows that such flow converges to free boundary embedded geodesic in infinite time, or shrinks to a round half-point on the boundary. As a consequence, we prove the existence of two free boundary embedded geodesics on a Riemannian -disk with a strictly convex boundary. Moreover, we prove that there exists a simple closed geodesic with Morse Index and . This settles the free boundary analog of Grayson's theorem.

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Submitted
Sep 18, 2023
Last revised
Nov 4, 2023 (v2)
Subjects
math.DG, math.AP

Comments: 23 pages, submitted version, fixed typos. Comments welcome!

Curve shorteningSingularities(Non)collapsingSurfacesFree boundary & capillary

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 Classification of convex ancient free boundary curve shortening flows in the disc (Theodora Bourni and Mat Langford, 2021)5Bourni 2021 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)7Chai 2022 A distance comparison principle for curve shortening flow with free boundary (Mat Langford and Jonathan J. Zhu, 2023)1Langford 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)4Bourni 2024 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)8Fujihara 2024 High Codimension Curve Shortening Flow with Free Boundary (Huy The Nguyen and Artemis A. Vogiatzi, 2026)6Nguyen 2026 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)2Bourni 2026 On the Gage-Hamilton-Grayson Theorem for free boundary curves (Yueheng Bao and Lorenzo Sarnataro, 2026)3Bao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A distance comparison principle for curve shortening flow with free boundary

    Mat Langford and Jonathan J. ZhuWider flows

    Topics Free boundary & capillary, Curve shortening, Singularities · Shared terms “chord arc”, “arc profile”, “reflected chord”

  2. Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows

    Topics Free boundary & capillary, Curve shortening, Singularities · Shared terms “half point”, “round half”, “finite time”

  3. On the Gage-Hamilton-Grayson Theorem for free boundary curves

    Yueheng Bao and Lorenzo SarnataroWider flows

    Topics Free boundary & capillary, Curve shortening, Singularities · Shared terms “Grayson Theorem”, “Langford Zhu”

  4. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows

    Topics Free boundary & capillary, Curve shortening

  5. Classification of convex ancient free boundary curve shortening flows in the disc

    Theodora Bourni and Mat LangfordWider flows

    Topics Free boundary & capillary, Curve shortening

  6. High Codimension Curve Shortening Flow with Free Boundary

    Huy The Nguyen and Artemis A. VogiatziWider flows

    Topics Free boundary & capillary, Curve shortening, Singularities · Shared terms “reflected”

  7. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows

    Topics Free boundary & capillary, Singularities · Shared terms “flow converges”, “finite time”

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

    Naotoshi FujiharaWider flows

    Topics Curve shortening, Surfaces, Singularities · 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 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.