Paper · Aug 11, 2022

On the evolution of hypersurfaces along their inverse spacetime mean curvature

Gerhard Huisken, Markus Wolff

math.DGWider flowsarXiv:2208.05709
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Abstract

We construct weak solutions for the evolution of hypersurfaces along their inverse space-time mean curvature in asymptotically flat maximal initial data sets. As the speed of the new flow is given by a space-time invariant, it can detect both future- and past-trapped apparent horizons. The weak solution extends concepts developed by Huisken-Ilmanen for inverse mean curvature flow and by Moore for inverse null mean curvature flow.

Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsBoundaries & non-compactPhysics

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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.

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    Conghan DongWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “apparent horizon”, “data set”, “asymptotically flat”

  2. The Penrose conjecture for initial data sets satisfying a -convexity condition

    Conghan DongWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “data set”, “apparent horizon”, “initial data”

  3. The Penrose inequality with charge for 2-convex initial data sets

    Tuan DolmenWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact · Shared terms “data set”, “apparent horizon”, “initial data”

  4. Hawking Mass Monotonicity for Initial Data Sets

    Sven HirschWider flows PDF

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “data set”, “initial data”

  5. Inverse mean curvature flow and Ricci-pinched three-manifolds

    Gerhard Huisken and Thomas Koerber PDF

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Same author Gerhard Huisken

  6. Inverse mean curvature flow with outer obstacle

    Kai XuWider flows PDF

    Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact

  7. Global weak solutions for the inverse mean curvature flow in the Heisenberg group

    Adriano Pisante and Eugenio VecchiWider flows PDF

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “Huisken”, “Ilmanen”

  8. 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 “spacetime mean”, “null mean”, “initial data”

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.