Paper · Dec 3, 2018

Mean Curvature Flow of Compact Spacelike Submanifolds in Higher Codimension

Brendan Guilfoyle, Wilhelm Klingenberg

math.DGWider flowsarXiv:1812.00710v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove long-time existence for mean curvature flow of a smooth -dimensional spacelike submanifold of an dimensional manifold whose metric satisfies the timelike curvature condition.

View PDF View HTML arXiv page
Submitted
Dec 3, 2018
Last revised
Jul 22, 2020 (v2)
Subjects
math.DG, math.AP
Journal
Trans. Amer. Math. Soc. 372.9 (2019) 6263-6281
DOI
10.1090/tran/7766

Comments: 19 pages LaTEX. Essentially this is Section 3 of the Proof of the Caratheodory Conjecture by the same authors, which is posted at arXiv:0808.0851

Mean curvature flowLong-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)8Lambert 2016 Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows (Brendan Guilfoyle and Wilhelm Klingenberg, 2017)5Guilfoyle 2017 Spacelike mean curvature flow (Ben Lambert and Jason D. Lotay, 2018)6Lambert 2018 Mean curvature flow in asymptotically flat product spacetimes (Klaus Kroencke, Oliver Lindblad Petersen, Felix Lubbe et al., 2019)4Kroencke 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 Parabolic evolution with boundary to the Bishop family of holomorphic discs (Brendan Guilfoyle and Wilhelm Klingenberg, 2023)3Guilfoyle 2023 Mean Curvature Flow in de Sitter space (Or Hershkovits and Leonardo Senatore, 2023)7Hershkovits 2023 High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension (Ben Andrews and Qiyu Zhou, 2026)1Andrews 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

    Ben Andrews and Qiyu ZhouWider flows

    Topics Physics · Shared terms “spacelike submanifold”, “dimensional spacelike”, “timelike”

  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 Wilhelm Klingenberg · Shared terms “spacelike”

  3. Parabolic evolution with boundary to the Bishop family of holomorphic discs

    Brendan Guilfoyle and Wilhelm KlingenbergWider flows

    Topics Mean curvature flow · Same authors Brendan Guilfoyle, Wilhelm Klingenberg

  4. 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”

  5. Evolving to Non-round Weingarten Spheres: Integer Linear Hopf Flows

    Brendan Guilfoyle and Wilhelm KlingenbergWider flows

    Same authors Brendan Guilfoyle, Wilhelm Klingenberg

  6. Spacelike mean curvature flow

    Ben Lambert and Jason D. Lotay

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

  7. Mean Curvature Flow in de Sitter space

    Or Hershkovits and Leonardo SenatoreWider flows

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

  8. A remark on gradient estimates for spacelike mean curvature flow with boundary conditions

    Ben LambertWider flows

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

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) 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.