Paper · Oct 7, 2016

A remark on gradient estimates for spacelike mean curvature flow with boundary conditions

Ben Lambert

math.DGWider flowsarXiv:1610.02202
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Abstract

We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension . This then implies that the mean curvature flow exists for all time and converges to a translating solution.

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Submitted
Oct 7, 2016
Subjects
math.DG
Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactFree boundary & capillaryLong-time existence & convergencePhysics

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 Mean Curvature Flows of Graphs with Neumann Boundary condition (Jinju Xu, 2016)3Xu 2016 Spacelike mean curvature flow (Ben Lambert and Jason D. Lotay, 2018)1Lambert 2018 Lagrangian mean curvature flow with boundary (Christopher G. Evans, Ben Lambert and Albert Wood, 2019)4Evans 2019 Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold Mⁿ×ℝ (Ya Gao, Yi-Juan Gong and Jing Mao, 2020)8Gao 2020 Level set mean curvature flow with Neumann boundary conditions (Satoru Aimi, 2021)5Aimi 2021 Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition (Jorge Lira and Fernanda Roing, 2022)6Lira 2022 A capillary problem for spacelike mean curvature flow in a cone of Minkowski space (Wilhelm Klingenberg, Ben Lambert and Julian Scheuer, 2023)2Klingenberg 2023 Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions (Can Cui and Nung Kwan Yip, 2025)7Cui 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Spacelike mean curvature flow

    Ben Lambert and Jason D. Lotay

    Topics Physics, Long-time existence & convergence, Boundaries & non-compact · Same author Ben Lambert · Shared terms “spacelike mean”, “boundary condition”, “Neumann”

  2. A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

    Wilhelm Klingenberg, Ben Lambert and Julian ScheuerWider flows

    Topics Free boundary & capillary, Physics · Same author Ben Lambert · Shared terms “spacelike mean”, “graphical”

  3. Mean Curvature Flows of Graphs with Neumann Boundary condition

    Jinju XuWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”, “Neumann boundary”, “gradient estimates”

  4. Lagrangian mean curvature flow with boundary

    Christopher G. Evans, Ben Lambert and Albert WoodWider flows

    Topics Free boundary & capillary, Self-shrinkers & translators, Long-time existence & convergence · Same author Ben Lambert · Shared terms “boundary condition”, “Neumann boundary”

  5. Level set mean curvature flow with Neumann boundary conditions

    Satoru AimiWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”, “Neumann boundary”

  6. Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition

    Jorge Lira and Fernanda RoingWider flows

    Topics Free boundary & capillary, Long-time existence & convergence, Boundaries & non-compact · Shared terms “boundary condition”, “Neumann boundary”

  7. Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

    Can Cui and Nung Kwan YipWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”, “Neumann boundary”, “gradient estimates”

  8. Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold

    Ya Gao, Yi-Juan Gong and Jing MaoWider flows

    Topics Free boundary & capillary, Self-shrinkers & translators · Shared terms “Neumann boundary”, “translating solutions”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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) 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. 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.