Paper · Jan 14, 2022

Graphical mean curvature flow with bounded bi-Ricci curvature

Renan Assimos, Andreas Savas-Halilaj, Knut Smoczyk

math.DGWider flowsarXiv:2201.05523v2Calculus of Variations and Partial Differential Equations 62, Article number: 12 (2023)
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Abstract

We consider the graphical mean curvature flow of strictly area decreasing maps , where is a compact Riemannian manifold of dimension and a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature of is bounded from below by the sectional curvature of . In addition, we obtain smooth convergence to a minimal map if . These results significantly improve known results on the graphical mean curvature flow in codimension .

Mean curvature flowLong-time existence & convergence

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Codimension two mean curvature flow of entire graphs

    Andreas Savas-Halilaj and Knut SmoczykWider flows PDF

    Topics Mean curvature flow · Same authors Andreas Savas-Halilaj, Knut Smoczyk · Shared terms “area decreasing”, “graphical mean”, “decreasing maps”

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

    Jingbo WanWider flows PDF

    Topics Mean curvature flow · Shared terms “graphical mean”, “area decreasing”

  3. The Genus-Decreasing Property of Mean Curvature Flow, I

    Brian WhiteWider flows PDF

    Topics Mean curvature flow · Shared terms “decreasing property”

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

    Ke Chen, Ruilin Hu and Quoc-Hung NguyenWider flows PDF

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

  5. Mean Curvature Flows of Two-Convex Lagrangians

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows PDF

    Topics Long-time existence & convergence · Shared terms “area decreasing”

  6. A New Monotone Quantity in Mean Curvature Flow Implying Sharp Homotopic Criteria

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows PDF

    Topics Mean curvature flow · Shared terms “graphical mean”, “area decreasing”, “Riemannian manifolds”

  7. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows PDF

    Topics Mean curvature flow · Shared terms “graphical mean”, “preserved”

  8. Mean Curvature Flow in de Sitter space

    Or Hershkovits and Leonardo SenatoreWider flows PDF

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

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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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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.