Paper · Oct 6, 2026

Radial Powered Mean Curvature Flow with Robin Boundary Conditions

Tianhong Pu, Xiaoqian Xin, Lixia Yuan

math.APWider flowsarXiv:2610.08283
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Abstract

We study a positive radial graph evolving by , with and , in the unit cylinder and with . We separate finite-time continuation estimates from the time-uniform interior estimates needed for convergence of normalized translates. On existing classical intervals we prove height and gradient bounds and an explicit conversion of velocity bounds into estimates for , , and . We derive the correct boundary defect for comparisons with finite-slope translators. Under independent interior velocity bounds and two-sided slope trapping, we prove compactness and identify every translated limit with the cup translator. We identify the radial zero-number comparison still required to establish that trapping. Stationary caps and power-law Robin laws are included.

Mean curvature flowSelf-shrinkers & translatorsSingularitiesLimit spaces & RCDBoundaries & non-compact

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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 Compactness and rigidity of self-shrinking surfaces (Tang-Kai Lee, 2021)7Lee 2021 No breather theorems for the mean curvature flow (Liang Cheng and Yongjia Zhang, 2021)8Cheng 2021 The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators (Brian White, 2022)5White 2022 Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles (Bendong Lou and Lixia Yuan, 2022)3Lou 2022 Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions (Xinfu Chen, Bendong Lou, Xiaoliu Wang et al., 2023)2Chen 2023 Uniqueness of Semigraphical Translators (Francisco Martín, Mariel Sáez and Raphael Tsiamis, 2023)4Martín 2023 Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2025)6Gama 2025 Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions (Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al., 2026)1Pu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

    Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al.Wider 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 Self-shrinkers & translators, Boundaries & non-compact · Same author Lixia Yuan · Shared terms “Robin boundary”, “boundary condition”, “zero number”

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

    Bendong Lou and Lixia YuanWider flows PDF

    Topics Self-shrinkers & translators · Same author Lixia Yuan

  4. Uniqueness of Semigraphical Translators

    Francisco Martín, Mariel Sáez and Raphael TsiamisWider flows PDF

    Topics Limit spaces & RCD, Self-shrinkers & translators · Shared terms “compactness”

  5. The Boundary Term in Huisken's Monotonicity Formula and the Entropy of Translators

    Brian WhiteWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact

  6. Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact, Singularities

  7. Compactness and rigidity of self-shrinking surfaces

    Tang-Kai LeeWider flows PDF

    Topics Limit spaces & RCD, Self-shrinkers & translators · Shared terms “compactness”

  8. No breather theorems for the mean curvature flow

    Liang Cheng and Yongjia ZhangWider flows PDF

    Topics Self-shrinkers & translators, Boundaries & non-compact

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.