Paper · May 28, 2022

Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles

Bendong Lou, Lixia Yuan

math.APWider flowsarXiv:2205.14431Journal de Mathematiques Pures et Appliquees, 176 (2023), 102-137
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Abstract

We study a generalized mean curvature flow involving a positive power of the mean curvature and a driving force. In this paper, we first construct all kinds of radially symmetric translating solutions, and then select one of them to satisfy a prescribed boundary angle in a cylinder. We then consider the flow starting at an initial hypersurface: showing the a priori estimates (especially the uniform-in-time bounds for the mean curvature which guarantee the uniform parabolicity of the corresponding fully nonlinear equation), giving the global existence for the solution of the initial boundary value problem, and proving its convergence to the corresponding translating solution. Our study provides a complete exposition on the influence of the dimension, the power of the mean curvature, the driving force and the boundary angles on the existence and stability of radially symmetric translating solutions.

Comments: 29 pages

Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessLong-time existence & convergenceRotational symmetry & warped products

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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 Rotationally symmetric translating solutions to extrinsic geometric flows (Sathyanarayanan Rengaswami, 2021)4Rengaswami 2021 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)1Chen 2023 Existence proofs for rotationally symmetric translating solutions to mean curvature flow (Hakar Raji and Oliver C. Schnürer, 2025)3Raji 2025 Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications (José Torres Santaella, 2025)6Santaella 2025 Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space (Ronaldo F. de Lima and Álvaro K. Ramos, 2026)7de Lima 2026 Translating solitons to higher order mean curvature flows in Riemannian products (Jorge Herbert Soares de Lira and Rafael Rocha de Farias, 2026)8de Lira 2026 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)5Davila 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions

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

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Long-time existence & convergence · Same authors Bendong Lou, Lixia Yuan · Shared terms “radially symmetric”

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

    Zhenghuan Gao, Bendong Lou and Jinju XuWider flows PDF

    Topics Self-shrinkers & translators · Same author Bendong Lou · Shared terms “translating solution”, “initial boundary”, “boundary value”

  3. Existence proofs for rotationally symmetric translating solutions to mean curvature flow

    Hakar Raji and Oliver C. SchnürerWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solution”, “symmetric translating”

  4. Rotationally symmetric translating solutions to extrinsic geometric flows

    Sathyanarayanan RengaswamiWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “translating solution”, “symmetric translating”

  5. A Bernstein problem for translating solutions to the mean curvature flow

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “translating solution”

  6. Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and Applications

    José Torres SantaellaWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness · Shared terms “symmetric translating”, “fully nonlinear”

  7. Dynamical Stability of Translating Solitons to Mean Curvature Flow in Hyperbolic Space

    Ronaldo F. de Lima and Álvaro K. RamosWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Stability & uniqueness

  8. Translating solitons to higher order mean curvature flows in Riemannian products

    Jorge Herbert Soares de Lira and Rafael Rocha de FariasWider flows PDF

    Topics Rotational symmetry & warped products, Self-shrinkers & translators

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.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.