Paper · May 29, 2018

Evolution of Contractions between Non-Compact Manifolds

Felix Lubbe

math.DGWider flowsarXiv:1805.12540
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Abstract

Let be a complete manifold with bounded geometry, such that for some positive constant . We investigate the mean curvature flow of the graphs of smooth length-decreasing maps . In this case, the solution exists for all times and the evolving submanifold stays the graph of a length-decreasing map . We further prove uniform decay estimates for all derivatives of order of along the flow.

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Submitted
May 29, 2018
Subjects
math.DG

Comments: 29 pages; comments are welcome! arXiv admin note: text overlap with arXiv:1608.05394, and with arXiv:1602.07595 by other authors

Mean curvature flowBoundaries & non-compactLong-time existence & convergence

Its neighbourhood

Connected papers

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)3Savas-Halila 2016 Shadows of graphical mean curvature flow (Wolfgang Maurer, 2016)8Maurer 2016 Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant (Xuezhang Chen, Pak Tung Ho and Liming Sun, 2016)5Chen 2016 Evolution of area-decreasing maps between two-dimensional Euclidean spaces (Felix Lubbe, 2016)1Lubbe 2016 Mean curvature flow in asymptotically flat product spacetimes (Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe et al., 2019)2Kroencke 2019 Graphical mean curvature flow with bounded bi-Ricci curvature (Renan Assimos, Andreas Savas-Halilaj and Knut Smoczyk, 2022)4Assimos 2022 Codimension two mean curvature flow of entire graphs (Andreas Savas-Halilaj and Knut Smoczyk, 2024)6Savas-Halila 2024 Well-posedness for the mean curvature flow on the half-space and on bounded domains (Ke Chen, Ruilin Hu and Quoc-Hung Nguyen, 2026)7Chen 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 area-decreasing maps between two-dimensional Euclidean spaces

    Felix LubbeWider flows

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

  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 Felix Lubbe

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

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Mean curvature flow · Shared terms “decreasing maps”, “bounded geometry”

  4. 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 “decreasing maps”, “bounded geometry”

  5. Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

    Xuezhang Chen, Pak Tung Ho and Liming SunWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact · Shared terms “Let”

  6. Codimension two mean curvature flow of entire graphs

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Boundaries & non-compact · Shared terms “decreasing maps”

  7. 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

  8. Shadows of graphical mean curvature flow

    Wolfgang MaurerWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact

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) 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. 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.