Paper · Apr 18, 2016

Shadows of graphical mean curvature flow

Wolfgang Maurer

math.DGWider flowsarXiv:1604.05238
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Abstract

We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in . If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of the flowing surface onto , the shadow of the flow. This moving shadow can be seen as a weak solution for mean curvature flow of hypersurfaces in with a Dirichlet boundary condition. Furthermore, we provide a lemma of independent interest to locally mollify the boundary of an intersection of two smooth open sets in a way that respects curvature conditions.

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Submitted
Apr 18, 2016
Subjects
math.DG

Comments: 12 pages, 1 figure

Mean curvature flowWeak & singular flowsBoundaries & non-compactLong-time existence & convergence

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 General curvature flow without singularities (Ling Xiao, 2016)4Xiao 2016 A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions (Yoshikazu Giga, Fumihiko Onoue and Keisuke Takasao, 2018)6Giga 2018 Mean curvature flow in asymptotically flat product spacetimes (Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe et al., 2019)2Kroencke 2019 Mean curvature flow of symmetric double graphs only develops singularities on the hyperplane of symmetry (Wolfgang Maurer, 2021)1Maurer 2021 Level set mean curvature flow with Neumann boundary conditions (Satoru Aimi, 2021)7Aimi 2021 Graphical mean curvature flow with bounded bi-Ricci curvature (Renan Assimos, Andreas Savas-Halilaj and Knut Smoczyk, 2022)8Assimos 2022 A level-set method for a mean curvature flow with a prescribed boundary (Xingzhi Bian, Yoshikazu Giga and Hiroyoshi Mitake, 2023)5Bian 2023 Well-posedness for the mean curvature flow on the half-space and on bounded domains (Ke Chen, Ruilin Hu and Quoc-Hung Nguyen, 2026)3Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow of symmetric double graphs only develops singularities on the hyperplane of symmetry

    Wolfgang MaurerWider flows

    Topics Weak & singular flows · Same author Wolfgang Maurer · Shared terms “graphical”

  2. Mean curvature flow in asymptotically flat product spacetimes

    Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe et al.Wider flows

    Topics Long-time existence & convergence, Boundaries & non-compact · Same author Wolfgang Maurer

  3. Well-posedness for the mean curvature flow on the half-space and on bounded domains

    Ke Chen, Ruilin Hu and Quoc-Hung NguyenWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact · Shared terms “Dirichlet boundary”, “graphical mean”, “boundary condition”

  4. General curvature flow without singularities

    Ling XiaoWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact · Shared terms “graphical mean”, “boundary condition”

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

    Xingzhi Bian, Yoshikazu Giga and Hiroyoshi MitakeWider flows

    Topics Weak & singular flows, Long-time existence & convergence · Shared terms “Dirichlet”

  6. A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions

    Yoshikazu Giga, Fumihiko Onoue and Keisuke TakasaoWider flows

    Topics Weak & singular flows, Boundaries & non-compact · Shared terms “boundary condition”, “Dirichlet”

  7. Level set mean curvature flow with Neumann boundary conditions

    Satoru AimiWider flows

    Topics Weak & singular flows, Boundaries & non-compact · Shared terms “boundary condition”

  8. Graphical mean curvature flow with bounded bi-Ricci curvature

    Renan Assimos, Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Long-time existence & convergence · Shared terms “graphical mean”

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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) 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)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  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.