Paper · Jul 26, 2019

Propagation of a Mean Curvature Flow in a Cone

Bendong Lou

math.DGWider flowsarXiv:1907.11520
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Abstract

We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being -periodic in its position. First, by constructing a family of self-similar solutions, we give a priori estimates for the radially symmetric solutions and prove the global existence. Then we consider the homogenization limit as , and use {\it the slowest self-similar solution} to characterize the solution, with error , in some finite time interval.

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Submitted
Jul 26, 2019
Subjects
math.DG
Mean curvature flowSingularitiesFree boundary & capillaryLong-time existence & convergenceRotational 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 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 flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces (Valentina-Mira Wheeler, 2016)8Wheeler 2016 Mean curvature flow with free boundary - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)6Wheeler 2016 Existence of self-similar solution of the inverse mean curvature flow (K. M. Hui, 2018)3Hui 2018 Singularity formation in axially symmetric mean curvature flow with Neumann boundary (John Head and Sevvandi Kandanaarachchi, 2019)5Head 2019 Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles (Bendong Lou and Lixia Yuan, 2022)1Lou 2022 A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder (Zhenghuan Gao, Bendong Lou and Jinju Xu, 2022)2Gao 2022 Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions (Xinfu Chen, Bendong Lou, Xiaoliu Wang et al., 2023)4Chen 2023 Existence of flat flows for volume-preserving mean curvature flow with contact angle (Jiwoong Jang, 2025)7Jang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles

    Bendong Lou and Lixia YuanWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence · Same author Bendong Lou · Shared terms “radially symmetric”, “priori estimates”, “global existence”

  2. A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder

    Zhenghuan Gao, Bendong Lou and Jinju XuWider flows

    Topics Free boundary & capillary · Same author Bendong Lou · Shared terms “contact angle”, “normal velocity”

  3. Existence of self-similar solution of the inverse mean curvature flow

    K. M. HuiWider flows

    Topics Rotational symmetry & warped products · Shared terms “symmetric solutions”, “radially symmetric”

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

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

    Topics Rotational symmetry & warped products, Long-time existence & convergence · Same author Bendong Lou · Shared terms “radially symmetric”

  5. Singularity formation in axially symmetric mean curvature flow with Neumann boundary

    John Head and Sevvandi KandanaarachchiWider flows

    Topics Free boundary & capillary, Rotational symmetry & warped products, Singularities

  6. Mean curvature flow with free boundary - Type 2 singularities

    Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary, Rotational symmetry & warped products, Singularities

  7. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows

    Topics Free boundary & capillary, Long-time existence & convergence, Singularities · Shared terms “contact angle”

  8. Mean curvature flow with free boundary in embedded cylinders or cones and uniqueness results for minimal hypersurfaces

    Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary, Long-time existence & convergence, Singularities · Shared terms “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.

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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.