Paper · Jan 6, 2017

Existence and non-existence of minimal graphs

Qi Ding, J. Jost, Y. L. Xin

math.DGWider flowsarXiv:1701.01674v2
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Abstract

We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded domains for a large class of prescribed boundary data. This result can be seen as a natural generalization of the classical sharp criterion for solvability of the minimal surface equation by Jenkins-Serrin. In contrast, we also construct a class of prescribed boundary data on just mean convex domains for which the Dirichlet problem in codimension 2 is not solvable. Moreover, we study existence and the uniqueness of minimal graphs by perturbation.

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Submitted
Jan 6, 2017
Last revised
May 17, 2022 (v2)
Subjects
math.DG, math.AP
Journal
J. Math. Pure Appl. 179 (2023), 391-424

Comments: 33 pages, comments are welcome

Mean curvature flowHomogeneous spacesStability & uniquenessMinimal & CMC surfaces

Its neighbourhood

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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 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 Dirichlet problem for f-minimal graphs (Jean-Baptiste Casteras, Esko Heinonen and Ilkka Holopainen, 2016)2Casteras 2016 Boundary regularity for mean curvature flows of higher codimension (Qi Ding, J. Jost and Y. L. Xin, 2017)4Ding 2017 Scherk-like Translators for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2019)8Hoffman 2019 On instability of Type (II) Lawson-Osserman Cones (Zhaohu Nie and Yongsheng Zhang, 2020)5Nie 2020 A level-set method for a mean curvature flow with a prescribed boundary (Xingzhi Bian, Yoshikazu Giga and Hiroyoshi Mitake, 2023)6Bian 2023 The Dirichlet problem for the minimal surface system on smooth domains (Caiyan Li and Hengyu Zhou, 2025)3Li 2025 An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow (Ari J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll et al., 2025)1Aiolfi 2025 Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in ℍ³ as Translation Surfaces (Tarcios Andrey Ferreira and João Paulo dos Santos, 2025)7Ferreira 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow

    Ari J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll et al.Wider flows

    Topics Minimal & CMC surfaces · Shared terms “boundary data”, “Jenkins Serrin”, “prescribed boundary”

  2. Dirichlet problem for -minimal graphs

    Jean-Baptiste Casteras, Esko Heinonen and Ilkka HolopainenWider flows

    Topics Minimal & CMC surfaces · Shared terms “minimal graphs”, “prescribed boundary”, “Dirichlet”

  3. The Dirichlet problem for the minimal surface system on smooth domains

    Caiyan Li and Hengyu ZhouWider flows

    Topics Minimal & CMC surfaces · Shared terms “surface system”, “Dirichlet”

  4. Boundary regularity for mean curvature flows of higher codimension

    Qi Ding, J. Jost and Y. L. XinWider flows

    Topics Mean curvature flow · Same authors Qi Ding, J. Jost, Y. L. Xin

  5. On instability of Type (II) Lawson-Osserman Cones

    Zhaohu Nie and Yongsheng ZhangWider flows

    Topics Minimal & CMC surfaces, Stability & uniqueness · Shared terms “minimal graphs”, “Dirichlet”

  6. A level-set method for a mean curvature flow with a prescribed boundary

    Xingzhi Bian, Yoshikazu Giga and Hiroyoshi MitakeWider flows

    Topics Homogeneous spaces · Shared terms “prescribed boundary”, “Dirichlet”, “convex domains”

  7. Classification of Minimal Surfaces and Solitons to the Mean Curvature Flow in as Translation Surfaces

    Tarcios Andrey Ferreira and João Paulo dos SantosWider flows

    Topics Minimal & CMC surfaces, Homogeneous spaces

  8. Scherk-like Translators for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows

    Topics Minimal & CMC surfaces, Stability & uniqueness

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) 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) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 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.