Paper · Jun 7, 2024

Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds

Luen-Fai Tam

math.DGWider flowsarXiv:2406.04667
PDF Abstract page

As on arXiv

Abstract

Motivated by previous study on mean curvature flow and prescribed mean curvature flow on spatially compact space or asymptotically flat spacetime, in this work we will find sufficient conditions for the short time existence of prescribed mean curvature flow on a Lorentz manifold with a smooth time function starting from a complete noncompact spacelike hypersurface. Long time existence and convergence will also be discussed. Results will be applied to study some prescribed mean curvature flows inside the future of the origin in the Minkowski spacetime. Examples of spacetime related to the existence and convergence results near the future null infinity of the Schwarzschild spacetime are also discussed.

Mean curvature flowBoundaries & non-compactLong-time existence & convergenceHarmonic maps & DeTurck trickPhysics

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 2021, 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 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)1Huang 2021 Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces (Giuseppe Gentile and Boris Vertman, 2022)2Gentile 2022 Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion (Helmut Abels, Felicitas Bürger and Harald Garcke, 2022)6Abels 2022 Entire convex curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)5Wang 2022 Entire σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)3Wang 2022 Rigidity of Lipschitz map using harmonic map heat flow (Man-Chun Lee and Luen-Fai Tam, 2022)7Lee 2022 Translating solutions and the entire Hessian curvature flow in Minkowski space (Qu Changzheng, Wang Zhizhang and Wo Weifeng, 2024)8Changzheng 2024 Foliation of constant harmonic mean curvature surfaces in asymptotic Schwarzschild spaces (Yaoting Gui, Yuqiao Li and Jun Sun, 2024)4Gui 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Same author Luen-Fai Tam · Shared terms “time existence”, “short time”, “complete noncompact”

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

    Giuseppe Gentile and Boris VertmanWider flows PDF

    Topics Physics, Boundaries & non-compact · Shared terms “prescribed mean”, “compact spaces”

  3. Entire curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows PDF

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

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

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

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

  5. Entire convex curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows PDF

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

  6. Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

    Helmut Abels, Felicitas Bürger and Harald GarckeWider flows PDF

    Topics Harmonic maps & DeTurck trick · Shared terms “time existence”, “short time”

  7. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick · Same author Luen-Fai Tam · Shared terms “Motivated”

  8. Translating solutions and the entire Hessian curvature flow in Minkowski space

    Qu Changzheng, Wang Zhizhang and Wo WeifengWider flows PDF

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

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. 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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.