Paper · Oct 6, 2026

Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

Tianhong Pu, Xiaoqian Xin, Lixia Yuan, Xiaoyu Zhang

math.APWider flowsarXiv:2610.08270
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Abstract

We study the graphical forced mean curvature flow in a planar strip subject to Robin boundary conditions. The normal velocity is prescribed by , where is constant and denotes the curvature of the graph. We consider positive even solutions and reduce the problem to the half interval. The main results describe the degeneration produced by the Robin condition: the height, the instantaneous velocity, and the spatial gradient diverge, while the normalized slope converges to one away from the symmetry axis. The normalized velocity converges to the forcing parameter . The curvature converges to zero in the interior in an integral sense, whereas its horizontal curvature measure concentrates at the symmetry axis.

Comments: 15 pages

Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsBoundaries & non-compact

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  1. Radial Powered Mean Curvature Flow with Robin Boundary Conditions

    Tianhong Pu, Xiaoqian Xin and Lixia YuanWider flows PDF

    Topics Boundaries & non-compact · Same authors Tianhong Pu, Xiaoqian Xin, Lixia Yuan · Shared terms “Robin boundary”, “boundary condition”, “velocity”

  2. Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions

    Xinfu Chen, Bendong Lou, Xiaoliu Wang et al.Wider flows PDF

    Topics Boundaries & non-compact · Same author Lixia Yuan · Shared terms “asymptotic dynamics”, “Robin boundary”, “boundary condition”

  3. Quenching for axisymmetric hypersurfaces under forced mean curvature flows

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    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  4. A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

    Jiwoong JangWider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”, “graphical”

  5. Level-set forced mean curvature flow with the Neumann boundary condition

    Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al.Wider flows PDF

    Topics Volume-preserving & constrained flows, Boundaries & non-compact · Shared terms “boundary condition”, “forced mean”

  6. The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus

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    Topics Volume-preserving & constrained flows, Other curvature flows

  7. Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals

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    Topics Volume-preserving & constrained flows, Boundaries & non-compact · Shared terms “forced mean”, “boundary condition”

  8. On a non-local area-preserving curve flow

    Zezhen Sun and Yuting WuWider flows PDF

    Topics Volume-preserving & constrained flows, Other curvature flows

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.