Paper · Mar 8, 2019

Mean curvature flow in asymptotically flat product spacetimes

Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe, Tobias Marxen, Wolfgang Maurer, Wolfgang Meiser, Oliver C. Schnürer, Áron Szabó, Boris Vertman

math.DGWider flowsarXiv:1903.03502v2
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Abstract

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold , where is asymptotically flat. If the initial hypersurface is uniformly spacelike and asymptotic to for some at infinity, we show that a mean curvature flow starting at exists for all times and converges uniformly to as .

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Submitted
Mar 8, 2019
Last revised
Aug 11, 2020 (v2)
Subjects
math.DG, math.AP
Journal
J. Geom. Anal (2020)
DOI
10.1007/s12220-020-00486-z

Comments: 23 pages, final version

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

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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. Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds

    Luen-Fai TamWider flows

    Topics Physics, Long-time existence & convergence, Boundaries & non-compact · Shared terms “asymptotically flat”, “spacelike hypersurface”

  2. Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces

    Giuseppe Gentile and Boris VertmanWider flows

    Topics Physics, Boundaries & non-compact · Same author Boris Vertman · Shared terms “Lorentzian”

  3. Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces

    Yaoting Gui, Yuqiao Li and Jun SunWider flows

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

  4. The Yamabe flow on asymptotically flat manifolds

    Eric Chen and Yi WangWider flows

    Topics Physics, Boundaries & non-compact · Shared terms “asymptotically flat”, “flow starting”

  5. Entire curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Physics, Long-time existence & convergence, Boundaries & non-compact · Shared terms “spacelike hypersurface”, “initial hypersurface”

  6. A volume-renormalized mass for asymptotically hyperbolic manifolds

    Mattias Dahl, Klaus Kroencke and Stephen McCormick

    Topics Physics, Boundaries & non-compact · Same author Klaus Kroencke

  7. Shadows of graphical mean curvature flow

    Wolfgang MaurerWider flows

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

  8. Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces

    Carlo Sinestrari and Jacopo TenanWider flows

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

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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

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.