Paper · Oct 25, 2025

Two dimensional anisotropic mean curvature flow with contact angle condition

Can Cui, Nung Kwan Yip

math.APWider flowsarXiv:2510.22136
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Abstract

In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in . We establish a prior gradient estimate for smooth solutions to this boundary value problem. The same approach can also handle Dirichlet boundary condition in , . For both problems, we prove that the solutions converge to one that is translation invariant in time.

Comments: 22 pages, 2 figures

Mean curvature flowAnisotropic & nonlocal flowsSurfacesBoundaries & non-compactFree boundary & capillary

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

    Can Cui and Nung Kwan YipWider flows PDF

    Topics Anisotropic & nonlocal flows, Free boundary & capillary, Boundaries & non-compact · Same authors Can Cui, Nung Kwan Yip · Shared terms “contact angle”, “boundary condition”, “translation invariant”

  2. A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

    Tokuhiro Eto and Yoshikazu GigaWider flows PDF

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

  3. Area-preserving curvature flow for open curves with general contact angles on skew lines

    Fuya HiroiWider flows PDF

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

  4. Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

    Kiichi TashiroWider flows PDF

    Topics Free boundary & capillary · Shared terms “contact angle”, “angle condition”

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

    Zhenghuan Gao, Bendong Lou and Jinju XuWider flows PDF

    Topics Free boundary & capillary · Shared terms “contact angle”, “boundary value”, “strictly convex”

  6. Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain

    Tokuhiro Eto and Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “contact angle”, “boundary condition”, “solution converges”

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

    Jiwoong JangWider flows PDF

    Topics Free boundary & capillary · Shared terms “contact angle”, “angle condition”

  8. Anisotropic Parabolic Obstacle Problems and the Stefan Problem: Regularity of the Evolving Free Boundary

    Ezequiel Barbosa, Rosivaldo Gonçalves and Luan de FigueiredoWider flows PDF

    Topics Anisotropic & nonlocal flows, Free boundary & capillary

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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.