Paper · Dec 23, 2024

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

Klaus Kroencke, Markus Wolff

math.DGWider flowsarXiv:2412.17563
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Abstract

We construct asymptotic foliations of asymtotically Schwarzschildean lightcones by surfaces of constant spacetime mean curvature (STCMC). Our construction is motivated by the approach of Huisken-Yau for the Riemannian setting in employing a geometric flow. We prove that initial data within a sufficient a-priori class converges exponentially to an STCMC surface under area preserving null mean curvature flow. Further, we show that the resulting STCMC surfaces form an asymptotic foliation that is unique within the a-priori class.

Comments: 75 pages

Mean curvature flowLong-time existence & convergence

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    Topics Mean curvature flow · Shared terms “constant spacetime”, “spacetime mean”, “STCMC”

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

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

  3. Volume preserving spacetime mean curvature flow and foliations of initial data sets

    Jacopo TenanWider flows PDF

    Topics Long-time existence & convergence · Shared terms “spacetime mean”, “foliation”, “Huisken Yau”

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

    Markus Wolff PDF

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

  5. Ricci flow on surfaces along the standard lightcone in the -Minkowski spacetime

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

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

    Markus WolffWider flows PDF

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

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

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

    Topics Long-time existence & convergence · Shared terms “foliation”, “surfaces form”, “initial data”

  8. Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space

    Yaoting Gui, Yuqiao Li and Jun SunWider flows PDF

    Topics Long-time existence & convergence · Shared terms “area preserving”, “converges exponentially”, “foliation”

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) 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.