Paper · Mar 20, 2024

The Riemannian Penrose Inequality with Matter Density

Hubert Bray, Yiyue Zhang

math.DGWider flowsarXiv:2403.13211
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As on arXiv

Abstract

Riemannian Penrose Inequalities are precise geometric statements that imply that the total mass of a zero second fundamental form slice of a spacetime is at least the mass contributed by the black holes, assuming that the spacetime has nonnegative matter density everywhere. In this paper, we remove this last assumption, and prove stronger statements that the total mass is at least the mass contributed by the black holes, plus a contribution coming from the matter density along the slice. We use the first author's conformal flow to achieve this, combined with Stern's harmonic level set techniques in the first case, and spinors in the second case. We then compare these new results to results previously known from Huisken-Ilmanen's inverse mean curvature flow techniques.

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Submitted
Mar 20, 2024
Subjects
math.DG

Comments: This paper was written for a special issue in memory of Robert Bartnik

Mean curvature flowOther curvature flowsInverse mean curvature flowGeometric inequalitiesPhysics

Its neighbourhood

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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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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Isoperimetric Riemannian Penrose Inequality

    Luca Benatti, Mattia Fogagnolo and Lorenzo MazzieriWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Riemannian Penrose”, “Penrose inequality”, “Huisken”

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

    Tuan DolmenWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Penrose inequality”, “Riemannian Penrose”

  3. Monotone Quantities for -Harmonic functions and the Sharp -Penrose inequality

    Liam Mazurowski and Xuan YaoWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Penrose inequality”

  4. On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes

    Brian Harvie and Ye-Kai WangWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics · Shared terms “Riemannian Penrose”, “black hole”, “Penrose inequality”

  5. Positive mass and isoperimetry for continuous metrics with nonnegative scalar curvature

    Gioacchino Antonelli, Mattia Fogagnolo, Stefano Nardulli et al.Wider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Physics

  6. The -inverse mean curvature flow and the generalized Penrose conjecture

    Conghan DongWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows · Shared terms “Penrose”

  7. Isoperimetry and the properness of weak inverse mean curvature flow

    Kai XuWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Huisken”, “Ilmanen”

  8. Hawking Mass Monotonicity for Initial Data Sets

    Sven HirschWider flows

    Topics Inverse mean curvature flow, Physics, Other curvature flows

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) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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.