Paper · Dec 12, 2023

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

Markus Wolff

math.DGarXiv:2312.07317
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Abstract

We consider null mean curvature flow along the standard lightcone in the de Sitter spacetime. This flow was first studied by Roesch–Scheuer along null hypersurfaces for the detection of MOTS, and independently by the author in the specific case of the standard Minkowski lightcone. Similar to the Minkowski case, null mean curvature flow along the de Sitter lightcone can be related to -Ricci flow for surfaces of genus by an appropriate rescaling. Building on this rescaling procedure, we analyse singularity formation, asymptotic behavior and ancient solutions to the flow.

Ricci flowMean curvature flowAncient solutionsSingularitiesPhysics

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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 On Ricci flows with closed and smooth tangent flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)5Chan 2021 Ricci flow on surfaces along the standard lightcone in the 3+1-Minkowski spacetime (Markus Wolff, 2022)1Wolff 2022 On the evolution of hypersurfaces along their inverse spacetime mean curvature (Gerhard Huisken and Markus Wolff, 2022)4Huisken 2022 A De Lellis-Müller type estimate on the Minkowski lightcone (Markus Wolff, 2023)2Wolff 2023 Mean Curvature Flow in de Sitter space (Or Hershkovits and Leonardo Senatore, 2023)7Hershkovits 2023 Revisiting generic mean curvature flow in ℝ³ (Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al., 2024)6Chodosh 2024 Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (Klaus Kroencke and Markus Wolff, 2024)3Kroencke 2024 A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II (Reto Buzano and Gianmichele Di Matteo, 2026)8Buzano 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow on surfaces along the standard lightcone in the -Minkowski spacetime

    Markus Wolff PDF

    Topics Singularities · Same author Markus Wolff · Shared terms “lightcone”, “null mean”, “along null”

  2. A De Lellis-Müller type estimate on the Minkowski lightcone

    Markus WolffWider flows PDF

    Topics Physics, Singularities · Same author Markus Wolff · Shared terms “lightcone”, “flow along”, “null mean”

  3. Foliations of asymptotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature

    Klaus Kroencke and Markus WolffWider flows PDF

    Topics Mean curvature flow · Same author Markus Wolff · Shared terms “lightcone”, “null mean”

  4. On the evolution of hypersurfaces along their inverse spacetime mean curvature

    Gerhard Huisken and Markus WolffWider flows PDF

    Topics Physics · Same author Markus Wolff · Shared terms “null mean”

  5. On Ricci flows with closed and smooth tangent flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics Ancient solutions, Singularities

  6. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows PDF

    Topics Ancient solutions, Singularities

  7. Mean Curvature Flow in de Sitter space

    Or Hershkovits and Leonardo SenatoreWider flows PDF

    Topics Physics · Shared terms “Sitter”

  8. A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature – Part II

    Reto Buzano and Gianmichele Di Matteo PDF

    Topics Ancient solutions, Singularities · Shared terms “Building”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.