Paper · Aug 6, 2018

Spacelike mean curvature flow

Ben Lambert, Jason D. Lotay

math.DGarXiv:1808.01994v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space , which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of the -Laplacian flow in cases related to coassociative fibrations.

View PDF View HTML arXiv page
Submitted
Aug 6, 2018
Last revised
Sep 2, 2019 (v2)
Subjects
math.DG
Journal
J. Geom. Anal. 31 (2021), 1291-1359
DOI
10.1007/s12220-019-00266-4

Comments: This is a pre-print of an article published in The Journal of Geometric Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s12220-019-00266-4

Mean curvature flowGeneralized & G₂ flowsBoundaries & non-compactLong-time existence & convergencePhysics

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 A remark on gradient estimates for spacelike mean curvature flow with boundary conditions (Ben Lambert, 2016)1Lambert 2016 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)7Lambert 2019 Lagrangian mean curvature flow with boundary (Christopher G. Evans, Ben Lambert and Albert Wood, 2019)8Evans 2019 A capillary problem for spacelike mean curvature flow in a cone of Minkowski space (Wilhelm Klingenberg, Ben Lambert and Julian Scheuer, 2023)2Klingenberg 2023 Long-time existence of some geometric flows with bounded scalar curvature (Chuanhuan Li and Yi Li, 2026)4Li 2026 Spacelike Brakke flows with boundary in pseudo-Euclidean space (Javier Hervás, 2026)3Hervás 2026 Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow (Tianyin Ren and Chong Song, 2026)5Ren 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A remark on gradient estimates for spacelike mean curvature flow with boundary conditions

    Ben LambertWider flows

    Topics Physics, Long-time existence & convergence, Boundaries & non-compact · Same author Ben Lambert · Shared terms “spacelike mean”, “boundary condition”, “Neumann”

  2. A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

    Wilhelm Klingenberg, Ben Lambert and Julian ScheuerWider flows

    Topics Physics · Same author Ben Lambert · Shared terms “spacelike mean”

  3. Spacelike Brakke flows with boundary in pseudo-Euclidean space

    Javier HervásWider flows

    Topics Physics · Shared terms “spacelike mean”, “pseudo Euclidean”, “Euclidean space”

  4. Long-time existence of some geometric flows with bounded scalar curvature

    Chuanhuan Li and Yi Li

    Topics Generalized & G₂ flows, Long-time existence & convergence · Shared terms “Laplacian flow”

  5. Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

    Tianyin Ren and Chong Song

    Topics Physics · Shared terms “spacelike mean”, “Euclidean space”

  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, Boundaries & non-compact · Shared terms “spacelike”

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

    Ben Lambert and Julian ScheuerWider flows

    Topics Physics · Same author Ben Lambert · Shared terms “spacelike”

  8. Lagrangian mean curvature flow with boundary

    Christopher G. Evans, Ben Lambert and Albert WoodWider flows

    Topics Long-time existence & convergence, Boundaries & non-compact · Same author Ben Lambert · Shared terms “boundary condition”, “Neumann”, “Dirichlet”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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.