Paper · Aug 18, 2016

Evolution of area-decreasing maps between two-dimensional Euclidean spaces

Felix Lubbe

math.DGWider flowsarXiv:1608.05394
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Abstract

We consider the mean curvature flow of the graph of a smooth map between two-dimensional Euclidean spaces. If satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map . Further, we prove uniform decay estimates for the mean curvature vector of the graph and all higher-order derivatives of the corresponding map .

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Submitted
Aug 18, 2016
Subjects
math.DG
DOI
10.1007/s12220-018-0006-6

Comments: 17 pages

Mean curvature flowSurfacesLong-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 Mean curvature flow of area decreasing maps between Riemann surfaces (Andreas Savas-Halilaj and Knut Smoczyk, 2016)2Savas-Halila 2016 Evolution of Contractions between Non-Compact Manifolds (Felix Lubbe, 2018)1Lubbe 2018 Mean curvature flow in asymptotically flat product spacetimes (Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe et al., 2019)8Kroencke 2019 Global transversal stability of Euclidean planes under skew mean curvature flow evolutions (Ze Li, 2020)4Li 2020 Graphical mean curvature flow with bounded bi-Ricci curvature (Renan Assimos, Andreas Savas-Halilaj and Knut Smoczyk, 2022)3Assimos 2022 Codimension two mean curvature flow of entire graphs (Andreas Savas-Halilaj and Knut Smoczyk, 2024)5Savas-Halila 2024 Sharp Interior Gradient Estimate for Area Decreasing Graphical Mean Curvature Flow in Arbitrary Codimension (Jingbo Wan, 2024)6Wan 2024 Calibration energy and mean curvature flow (Tatsuya Miura and Fabian Rupp, 2026)7Miura 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Evolution of Contractions between Non-Compact Manifolds

    Felix LubbeWider flows

    Topics Long-time existence & convergence · Same author Felix Lubbe · Shared terms “decreasing maps”, “submanifold stays”, “uniform decay”

  2. Mean curvature flow of area decreasing maps between Riemann surfaces

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Surfaces · Shared terms “decreasing maps”, “area decreasing”

  3. 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 “area decreasing”, “decreasing maps”, “decreasing property”

  4. Global transversal stability of Euclidean planes under skew mean curvature flow evolutions

    Ze LiWider flows

    Topics Surfaces, Long-time existence & convergence · Shared terms “dimensional Euclidean”

  5. Codimension two mean curvature flow of entire graphs

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Mean curvature flow · Shared terms “area decreasing”, “decreasing maps”, “decreasing property”

  6. Sharp Interior Gradient Estimate for Area Decreasing Graphical Mean Curvature Flow in Arbitrary Codimension

    Jingbo WanWider flows

    Topics Mean curvature flow · Shared terms “area decreasing”

  7. Calibration energy and mean curvature flow

    Tatsuya Miura and Fabian RuppWider flows

    Topics Surfaces, Long-time existence & convergence · Shared terms “Euclidean space”

  8. Mean curvature flow in asymptotically flat product spacetimes

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

    Topics Long-time existence & convergence · Same author Felix Lubbe

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) 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) 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. 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.