Paper · Mar 30, 2021

Mean curvature flow in null hypersurfaces and the detection of MOTS

Henri Roesch, Julian Scheuer

math.DGWider flowsarXiv:2103.16402
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As on arXiv

Abstract

We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an un-trapped foliation asymptotically.

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Submitted
Mar 30, 2021
Subjects
math.DG, math-ph, math.AP
Journal
Commun. Math. Phys. 390, no. 3, (2022), p. 1149-1173
DOI
10.1007/s00220-022-04326-9

Comments: 24 pages

Mean curvature flow3-manifoldsLong-time existence & convergencePhysics

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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 Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature (Heiko Kröner and Julian Scheuer, 2017)8Kröner 2017 Mean curvature flow in asymptotically flat product spacetimes (Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe et al., 2019)6Kroencke 2019 Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces (Ben Lambert and Julian Scheuer, 2019)3Lambert 2019 Parabolic approaches to curvature equations (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2020)4Bryan 2020 The quermassintegral preserving mean curvature flow in the sphere (Esther Cabezas-Rivas and Julian Scheuer, 2022)2Cabezas-Riva 2022 A capillary problem for spacelike mean curvature flow in a cone of Minkowski space (Wilhelm Klingenberg, Ben Lambert and Julian Scheuer, 2023)1Klingenberg 2023 An area growth argument for null mean curvature flow along the standard de Sitter lightcone (Markus Wolff, 2023)5Wolff 2023 Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)7Gui 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

    Wilhelm Klingenberg, Ben Lambert and Julian ScheuerWider flows

    Topics Physics, 3-manifolds · Same author Julian Scheuer · Shared terms “spacelike”

  2. The quermassintegral preserving mean curvature flow in the sphere

    Esther Cabezas-Rivas and Julian ScheuerWider flows

    Topics Physics, Long-time existence & convergence · Same author Julian Scheuer · Shared terms “converges smoothly”, “flow exists”

  3. Isoperimetric problems for spacelike domains in generalized Robertson-Walker spaces

    Ben Lambert and Julian ScheuerWider flows

    Topics Physics · Same author Julian Scheuer · Shared terms “spacelike”

  4. Parabolic approaches to curvature equations

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Physics · Same author Julian Scheuer · Shared terms “resembles”, “spacelike”, “Euclidean space”

  5. An area growth argument for null mean curvature flow along the standard de Sitter lightcone

    Markus Wolff

    Topics Physics · Shared terms “null hypersurface”, “MOTS”, “detection”

  6. Mean curvature flow in asymptotically flat product spacetimes

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

    Topics Physics, Long-time existence & convergence · Shared terms “spacelike”

  7. 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 · Shared terms “foliation”, “initial surface”

  8. Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

    Heiko Kröner and Julian ScheuerWider flows

    Topics Long-time existence & convergence · Same author Julian Scheuer · Shared terms “converges smoothly”, “Euclidean”

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)

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    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

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    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.