Paper · Jun 29, 2025

The Dirichlet problem for the minimal surface system on smooth domains

Caiyan Li, Hengyu Zhou

math.DGWider flowsarXiv:2506.23245
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Abstract

In this paper, we propose a new assumption (1.2) that involves a small oscillation and norms for maps from smooth bounded domains into Euclidean spaces. Furthermore, by assuming that the domain has non-negative Ricci curvature, we establish the Dirichlet problem for the minimal surface system via the mean curvature flow (MCF) with boundary. The long-time existence of such flow is derived using Bernstein-type theorems of higher codimensional self-shrinkers in the whole space and the half-space. Another novel aspect is that our hypothesis imposes no restriction on the diameter of the domains, which implies an existence result for an exterior Dirichlet problem of the minimal surface system.

Comments: 27 pages, Comments are welcome

Mean curvature flowSelf-shrinkers & translatorsLong-time existence & convergenceMinimal & CMC surfaces

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    Topics Minimal & CMC surfaces, Self-shrinkers & translators · Shared terms “half space”

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    Topics Minimal & CMC surfaces · Shared terms “Dirichlet”, “restriction”, “Euclidean space”

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    Topics Minimal & CMC surfaces · Shared terms “Bernstein type”, “type theorem”, “half space”

  4. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “long time”

  5. Invariant -translators in the Heisenberg group

    Antonio BuenoWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators

  6. Finite entropy translating solitons in slabs

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

    Topics Minimal & CMC surfaces, Self-shrinkers & translators

  7. The global structure of bi-rotational self-shrinkers

    Gregório Silva NetoWider flows PDF

    Topics Minimal & CMC surfaces, Self-shrinkers & translators

  8. Minimal hypersurfaces in by doubling the equatorial

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    Topics Minimal & CMC surfaces, 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) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.