Paper · Jul 14, 2020

Recent rigidity results for graphs with prescribed mean curvature

Bruno Bianchini, Giulio Colombo, Marco Magliaro, Luciano Mari, Patrizia Pucci, Marco Rigoli

math.DGWider flowsarXiv:2007.07194v2
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Abstract

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs . Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain , namely, for CMC graphs satisfying an overdetermined boundary condition.

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Submitted
Jul 14, 2020
Last revised
Nov 15, 2020 (v2)
Subjects
math.DG, math.AP
Journal
Mathematics in Engineering, 2021, 3(5): 1-48
DOI
10.3934/mine.2021039

Comments: 48 pages. Final version

Mean curvature flowStability & uniquenessBoundaries & non-compactFree boundary & capillaryRotational symmetry & warped productsDiscrete & graph Ricci

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 Mean Curvature Flows of Graphs with Neumann Boundary condition (Jinju Xu, 2016)3Xu 2016 Mean curvature flow solitons in the presence of conformal vector fields (Luis Alías, Jorge H. de Lira and Marco Rigoli, 2017)8Alías 2017 Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds (Zheng Huang, Zhou Zhang and Hengyu Zhou, 2017)4Huang 2017 Remarks on mean curvature flow solitons in warped products (Giulio Colombo, Luciano Mari and Marco Rigoli, 2018)1Colombo 2018 On the 1/H-flow by p-Laplace approximation: new estimates via fake distances under Ricci lower bounds (Luciano Mari, Marco Rigoli and Alberto Giulio Setti, 2019)5Mari 2019 Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain (Jiwoong Jang, 2021)6Jang 2021 On mean curvature flow solitons in the sphere (Marco Magliaro, Luciano Mari, Fernanda Roing et al., 2025)2Magliaro 2025 A Bernstein problem for translating solutions to the mean curvature flow (Juan Davila, Manuel del Pino and Juncheng Wei, 2026)7Davila 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Remarks on mean curvature flow solitons in warped products

    Giulio Colombo, Luciano Mari and Marco RigoliWider flows

    Topics Rotational symmetry & warped products, Stability & uniqueness · Same authors Giulio Colombo, Luciano Mari, Marco Rigoli · Shared terms “emphasis”, “splitting”

  2. On mean curvature flow solitons in the sphere

    Marco Magliaro, Luciano Mari, Fernanda Roing et al.Wider flows

    Topics Rotational symmetry & warped products · Same authors Marco Magliaro, Luciano Mari

  3. Mean Curvature Flows of Graphs with Neumann Boundary condition

    Jinju XuWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”, “gradient estimates”, “maximum principle”

  4. Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds

    Zheng Huang, Zhou Zhang and Hengyu ZhouWider flows

    Topics Discrete & graph Ricci, Rotational symmetry & warped products · Shared terms “ambient space”

  5. On the -flow by -Laplace approximation: new estimates via fake distances under Ricci lower bounds

    Luciano Mari, Marco Rigoli and Alberto Giulio SettiWider flows

    Topics Mean curvature flow · Same authors Luciano Mari, Marco Rigoli

  6. Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain

    Jiwoong JangWider flows

    Topics Free boundary & capillary, Rotational symmetry & warped products, Boundaries & non-compact · Shared terms “prescribed mean”, “gradient estimates”, “boundary condition”

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

    Juan Davila, Manuel del Pino and Juncheng WeiWider flows

    Topics Rotational symmetry & warped products, Stability & uniqueness · Shared terms “minimal graphs”, “entire minimal”, “detailed analysis”

  8. Mean curvature flow solitons in the presence of conformal vector fields

    Luis Alías, Jorge H. de Lira and Marco RigoliWider flows

    Topics Rotational symmetry & warped products · Same author Marco Rigoli

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) 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) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  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.