Paper · Jan 4, 2020

Contracting convex surfaces by mean curvature flow with free boundary on convex barriers

Sven Hirsch, Martin Li

math.APWider flowsarXiv:2001.01111
View PDF View HTML arXiv page

As on arXiv

Abstract

We consider the mean curvature flow of compact convex surfaces in Euclidean -space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite time. Moreover, the solution is asymptotic to a shrinking half-sphere lying in a half space. This extends, in dimension two, the convergence result of Stahl for umbilic barriers to general convex barriers. We introduce a new perturbation argument to establish fundamental convexity and pinching estimates for the flow. Our result can be compared to a celebrated convergence theorem of Huisken for mean curvature flow of convex hypersurfaces in Riemannian manifolds.

View PDF View HTML arXiv page
Submitted
Jan 4, 2020
Subjects
math.AP, math.DG

Comments: 32 pages, 1 figure

Mean curvature flowSingularitiesCurvature pinchingFree 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 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 free-boundary Brakke flow (Nick Edelen, 2016)1Edelen 2016 Mean convex mean curvature flow with free boundary (Nick Edelen, Robert Haslhofer, Mohammad N. Ivaki et al., 2019)3Edelen 2019 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)2Chai 2022 Classification of convex ancient free boundary mean curvature flows in the ball (Theodora Bourni and Mat Langford, 2022)8Bourni 2022 Free boundary flow with surgery (Robert Haslhofer, 2023)5Haslhofer 2023 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)7Bourni 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)6Bao 2026 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)4Ghosh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The free-boundary Brakke flow

    Nick EdelenWider flows

    Topics Free boundary & capillary · Shared terms “barrier surface”

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

    Xiaoxiang ChaiWider flows

    Topics Free boundary & capillary, Singularities · Shared terms “convex hypersurfaces”, “finite time”

  3. Mean convex mean curvature flow with free boundary

    Nick Edelen, Robert Haslhofer, Mohammad N. Ivaki et al.Wider flows

    Topics Free boundary & capillary, Singularities

  4. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows

    Topics Free boundary & capillary, Singularities

  5. Free boundary flow with surgery

    Robert HaslhoferWider flows

    Topics Free boundary & capillary, Singularities · Shared terms “convex surfaces”, “sufficiently”, “finite time”

  6. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows

    Topics Free boundary & capillary, Singularities

  7. 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, Singularities · Shared terms “finite time”

  8. Classification of convex ancient free boundary mean curvature flows in the ball

    Theodora Bourni and Mat LangfordWider flows

    Topics Free boundary & capillary

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

  1. 12B.6
    Bounded Curvature at Bounded DistanceBook 12B · κ-Solutions and Surgery

    The heart of the canonical neighbourhood proof: why curvature at bounded rescaled distance from a point of high curvature stays bounded, via incomplete limits, cones and Hamilton–Ivey pinching (Morgan–Tian, chapter 10).

  2. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  3. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.